Stochastic Analysis - Classical And Quantum - T. Hida (World, 2005) WW

Stochastic Analysis: h Classical and Quantum P u s p u t i v u of W h i t a

Noist

Theory

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I:t ochastic Analysis: Classical and Quantum P u s p u t i v u of W h i t 4 N o i s 4 Theory

Meijo University, Nagoya, Japan

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STOCHASTIC ANALYSIS: CLASSICAL AND QUANTUM Perspectives of White Noise Theory Copyright 0 2005 by World Scientific Publishing Co. Re. Ltd.

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Preface

We felt that time has come for a new epoch in stochastic analysis. Indeed various fields in mathematics as well as related fields in science continue to crossfertilize each other, while keeping good relationships with probability theory. There, dominant roles have been played by stochastic analysis (classical and quantum). It therefore seems to be a good opportunity to organize a conference on important topics in stochastic analysis. Almost three decades has passed since white noise analysis was launched, we thus plan to have perspectives of the theory on this occasion. Consequently, the conference “Stochastic Analysis: Classical and Quantum - Perspectives of White Noise Theory” took place at Meijo University, Nagoya, Japan for the period of November 1-5, 2004. The organizers of the conference were extremely happy to see many eminent mathematicians having contributed to the success of the conference and cultivated new ideas. To our great pleasure, important papers presented at the conference are published in the Proceedings of the conference. As such, we are grateful to the respective authors and to the referees of those papers. We acknowledge gratefully the general support of Meijo University and Ministry of Education, Culture, Sports, Science and Technology for the conference. Special thanks are also due to Professors M. Rockner, L. Streit and T. Shimizu who gave financial support together with me for the publication of this Proceedings. Finally, I wish to note the great help given by members of the local organizing committee: Professors M. Hitsuda, S. Ihara, K. Saito and Si Si. In particular, it is to be mentioned that this Proceedings would not appear without the help of Professor Si Si who handled the aspect of editing. July 2005

Takeyuki Hida

V

Organizing Committee of the Conference

Luigi Accardi Takeyuki Hida Hui-Hsiung Kuo Masanori Ohya Michael Rockner Ludwig Streit

vi

Contents

Preface

V

Part I White Noise Functional Approach to Polymer Entanglements C. C. Bernido and M . V. Carpio-Bernido

1

White Noise Analysis, Quantum Field Theory, and Topology

A . Hahn

13

A Topic on Noncanonical Representations of Gaussian Processes Y , Hibino

31

Integral Representation of Hilbert-Schmidt Operators on Boson Fock Space

U. C. Ji

35

The Dawn of White Noise Analysis I. Kubo

46

White Noise Stochastic Integration

57

H.-H. K u o Connes-Hida Calculus and Bismut-Quillen Superconnections

72

R. Le'andre and H. Ouerdiane A Quantum Decomposition of L6vy Processes Y.-J. Lee and H.-H. Shih

86

Generalized Entanglement and its Classification 100

T. Matsuoka A White Noise Approach to Fractional Brownian Motion D. Nualart

vii

112

Adaptive Dynamics in Quantum Information and Chaos M.Ohya

127

Micro-Macro Duality in Quantum Physics I. Ojima

143

White Noise Measures Associated to the Solutions of Stochastic Differential Equations H. Ouerdiane

162

A Remark on Sets in Infinite Dimensional Spaces with Full or Zero Capacity J . Ren and M . Rockner

177

An Infinite Dimensional Laplacian in White Noise Theory K. Saitd

187

Invariance of Poisson Noise Si Si, A. Tsoi and Win Win Htay

199

Nonequilibrium Steady States with Bose-Einstein Condensates S. Tasaki and T. Matsui

211

Multidimensional Skew Reflected Diffusions G. Trutnau

228

On Quantum Mutual Type Entropies and Quantum Capacity N . Watanabe

245

Part I1 White Noise Calculus and Stochastic Calculus L. Accardi and A. Boukas

viii

260

White Noise Functional Approach to Polymer Entanglements Christopher C. Bernido and M. Victoria Carpio-Bernido’ Research Center for Theoretical Physics Central Visayan Institute Foundation Jagna, Bohol 6308, Philappines Abstract The Hida-Streit white noise path integral is used t o investigate the entanglement probabilities of two chainlike macromolecules where one polymer lies on a plane and the other perpendicular to it. To simulate the data contained in the lineal structure of a polymer of length L which lies on the plane, a potential, V = f(s) ‘19, is introduced where, f = df/ds, 0 5 s 5 L , and f(s) a modulating function. Using the Ttransform in white noise calculus, entanglement probabilities are calculated which show a significant influence of chirality or the “handedness” of the polymer. The freedom to choose the modulating function f ( s ) , which gives rise to different entanglement probabilities, allows one to control and predict the coiling behavior of polymers. As examples, we consider two cases: (a) f (s) = kcos (vs),and (b) f (s) = ksp.

1

Introduction

Investigations in biochemistry reveal that protein molecules are able to carry out their biological functions only when they are folded into specific three-dimensional structures [l]. For instance, enzymes, which are essentially protein molecules, have highly specific shapes which allow them to receive their targets as a lock receives a key. Understanding this molecular recognition process, which depends on the structure of proteins, acquires importance since almost all chemical processes within a living organism rely on enzyme catalysis. What are the rules involved in forming protein structures? What are the factors which determine the manner in which proteins fold? One essential factor which has been identified is the one-dimensional sequence of data embodied in the repeating units of macromolecules. It has been noted that the genetic code is translated from DNA sequences to amino acid sequences, and this ‘Electronic mail: [email protected]

1

one-dimensional sequence of data influences the highly specific shapes of proteins. Moreover, chirality, or “handedness,” of macromolecules also plays an important role in the globular structure of proteins. It is known, for instance, that amino acids in proteins are “left-handed,” and that the chirality of amino acids manifests in the helical structures of the proteins they form. These observations give rise to more specific questions. How can the sequence of data in the lineal structure of macromolecules allow us to predict or determine a protein’s threedimensional structure? To what extent does chirality influence the folding or unfolding of proteins? In this paper, we present a simpIe model which may shed light into these questions. In particular, we look at a model which incorporates the following features observed in a macromolecule: (1)the ability of a macromolecule to use the one-dimensional sequence of data in its repeating units to influence its globular structure, and,

(2) the chirality of the polymer which influences the threedimensional structure of a macromolecule. We start with a polymer entanglement scenario originally studied by Edwards [2] and Prager and Frisch [3]. We then extend this system [4] by simulating the data contained in the repeating units of the entangled polymer using a potential of the form, V = f(s) 6, where f = d f / d s , and 19 is an angular variable about the z-axis. Here, f (s) is a modulating function where, 0 5 s 5 L , and L is the length of the polymer. We shall see that for any modulating function f (s), the “handedness” of the winding polymer has a significant effect on the entanglement probabilities. In particular, we look at two cases (a) f(s) = kcos(vs), and (b) f (s) = ksp, where k is a positive constant and p = f l ,f 2 , f 3 , .... Our calculations are greatly facilitated by first parametrizing the probability function in terms of the white noise variable, as was done by Hida and Streit [5] for the case of quantum propagators. The white noise functional can then be evaluated in a straightforward manner using the T-transform of white noise calculus [7]-[9].

2

Two Chainlike Macromolecules

In 1967, S. F. Edwards [2] and, independently, S. Prager and H. L. Frisch [3], solved the entanglement problem of two chainlike macromolecules in the absence of intermolecular forces. The problem consists of a polymer on a plane whose motion is constrained by a straight polymer orthogonal to the plane, since the macromolecules cannot cross each other. The polymer on the plane which starts at ro and ends at rl has fixed 2

endpoints, and can be viewed as a random walk with paths that can entangle, clockwise or counterclockwise, around the straight polymer which intersects the origin of the plane. Employing polar coordinates r =(r,6) for this problem, S. F. Edwards [2] used the Wiener representation of the random walk in which the probability is represented by,

where the integral is taken over all paths r(s) such that r(0) = ro and r(L) = rl. Here, we represent the polymer of length L as consisting of N freely hinged individual molecules, each of length 1 such that L = N1. In view of the point singularity, a set of topologically equivalent configurations can be characterized by a winding number n, where n = 0, k l , f2,..., indicating the number of times the polymer turns around the straight polymer intersecting the plane at the origin ( n 2 0, signifies n turns counterclockwise, and n 5 -1 means (n+ 11 turns clockwise).

3

Entanglement with an Intermolecular Potential

v (4 In 1977, F. W. Wiegel [S]extended this entanglement problem to include an intermolecular force where the repeating units of the entangled polymer interact with the straight polymer. For any potential V ( r )which has a minimum at some radius R, Wiegel obtained a low-temperature limit for the entanglement probabilities given by,

W ( n )= ( R / l ) W e x p (-47r2n2R2/N12) ;

( N >> 1).

(3.1)

For example, the potential of the form, V = C r 2 +D /r 2(C > 0, D > 0), was considered where R = (D/C)'I4 is a radius where the potential has a minimum. With this potential, the force is repulsive at short distances and attractive at large distances. Wiegel then obtained the entanglement probabilities for this harmonically bound polymer to be that of Eq. (3.1). For low temperatures, he also noted that the configurations of the polymer are confined to a narrow strip in the immediate vicinity of a circle around the origin with radius R . Below, we shall use these observations of Wiegel which we refer to as the generic case.

4

White Noise Path Integral Approach

Let us now familiarize ourselves with the white noise path integral approach by using it to arrive at Eq. (3.1). Since we are interested in 3

the number of possible windings around the origin that the polymer on the plane undergoes, we can simplify the calculation by fixing the radial variable to T = R, i.e., r =(R,6 ) ,and use 6 to track the number of turns, clockwise or counterclockwise, around the origin. As mentioned in the previous section, a fixed radial part describes the entanglement scenario in the low temperature limit [6] for any polymer interaction potential V ( T which ) has a minimum at some value T = R. For the generic case, Eq. (2.1) reduces to, L

P(61,60)= / e x p

[ - + / R 2 ($)'ds]

'D[Rdd],

(4.1)

0

with, d1 = 6 ( L ) and 60= d(0). The paths 6 can be parametrized as,

6 ( L )=60

+ ( h / R ) B(L) L

=60

+ (&/R) / w ( s )

ds,

(4.2)

0

where B ( s )is a Brownian motion parametrized by s, and w ( s ) a random white noise variable. With Eq. (4.2), the integrand in P(61,60)becomes,

Noting that the polymer can wind n times, clockwise or counterclockwise, we use the Donsker delta function

to fix the endpoint 61, where n = 0, f l ,5 2 , ... . Since P(t91,60) is now expressed as a white noise functional, the integration over D [ R dt9] becomes an integration over, Nu d"w = exp [(1/2) Sw(s)' ds] dp(w), where dp(w) is the Gaussian white noise measure. Eq. (4.1) can now be written as,

where

4

10 = Nexp

(-:

)w(s).

ds) .

The evaluation of P(61,60)is facilitated by using the Fourier representation of the 6 - function, i.e.,

Observing that the integration over dp(w) is just the T-transform of 10 [7]-[9],we obtain,

x exp (-X2ZL/4R2) dX

n=-w

Here, the Pn is the probability function for polymer configurations which entangle n - times around the origin. The remaining integral in Pn is a Gaussian integral over A. We have,

=m

e

x

p

[- (R2/1L)(60- 61 + 27rn)2] .

Also, applying Poisson's sum formula,

to Eq. (4.8),we get,

5

(4.9)

x exp [ i X (60- 61) - X2(ZL/4R2)] dX =

1 -

+O0

27r m=--00

exp [-im(60- dl) - m2(1L/4R2)]. (4.11)

For an arbitrary initial starting point we may set, 60 = 61, and the probability that the polymer winds n - times is,

-

d m e x p [- (27rn~)’ /ZL] +m

2?r

C

(4.12)

exp [-m2(1L/4R2)]

m=-m

For a very long polymer, L = N1 >> 1, the dominant term in the denominator is for m = 0. Hence,

W ( n )= ( R / l ) W e x p (-47r2n2R2/N12);

( N >> l),

(4.13)

which agrees with the result, Eq. (3.1), obtained by Wiegel [6].

5

Length-dependent Potentials

We shall now generalize the system discussed in the previous section by adding a length-dependent potential, V = f(s)6, acting on the polymer on the plane as it entangles around the second straight polymer at the origin [4]. Here, f = df /ds, where f (s) is the modulating function. The potential V is added to the “kinetic part” of Eq. (4.1) such that the probability function becomes,

(5.1) The nature of the potential may be understood in the following way. Firstly, one may associate with it a length-dependent force given by,

F = -VV = - f ( s ) / R . Secondly, the effect of the potential term may also be understood by rewriting it as,

6

L

= f ( L ) S ( L )- f ( 0 ) 6 ( 0 ) -

1fS

ds

.

(5.2)

0

The first two terms are constants given by the values of f and 6 at the endpoints. The last term, on the other hand, shows that one essentially has a “velocity-dependent potential” in view of

s. Moreover, from Eq.

(5.2), one may have the case, f(s)= 0 with f # 0, such that the nonzero f may still manifest in the probability function. For the case, f = 0, one obtains the results of the generic case discussed in the previous section. An example of a constant nonzero f may be illustrated if one takes, f = (q@0/27r),where q is the net charge of each repeating unit of the polymer which winds around the straight polymer that contains a constant magnetic flux @O oriented along the z-axis. This choice leads to an effective potential, f $ = qA . i, which resembles that of an AharonovBohm setup where A is the vector potential for the constant magnetic flux @o [4,101. Using again the parametrization Eq. (4.2), we obtain an expression similar to Eq. (4.7) but modified by the potential Eq. (5.2) of the form,

L

x Jexp { i s

( d i / ~(X) - if

) w ( s ) ds

0

XIO dp(w) dX

.

(5.3)

The integration over dp(w) is again just the T - transform of I0 which yields,

7

c +m

=

Pn.

(5.4)

The Gaussian integral over X in Pn can be evaluated to give,

Employing the Poisson sum formula to Eq. (5.4), and integrating X yields, +m

L

m21L 4R2

L

d s + 1s $ f ' d s } .

iml 2R2

(5.6)

0

From these, we obtain the probability that the polymer entangles n times as, (setting, 60 = &),

8

where &(u) is the theta function [ll], +W

e3(4= 1+ 2 C qmz cos(2mu), m=l

with u = (1/4R2)j” f ds, and q = exp (-N12/4R2).

6

Chirality of Entangled Polymers

Let us now consider the effect of the “handedness” of a polymer on the coiling probabilities of a macromolecule. As is normally the case, we define “handedness” in a way that a “right-handed” polymer would have a mirror-image which is “left-handed.” We may write Eq. (5.7) as,

where W,, symmetric in n, is of the form,

From these equations the following observations may be made: (1) It is clear from Eq. (6.1) that the entanglement probability W ( n )significantly changes depending on whether n is a positive or a negative number. If we designate clockwise winding ( n -1) as “right-handed,” and anti-clockwise winding ( n 2 0) as “left-handed,” then for f > 0, a “righthanded” polymer is more likely to have configurations with large values of winding number n than “left-handed” ones. In particular, for winding numbers k n , the corresponding “right” and “left-handed” entanglements differ by an expcnential factor, i.e., W ( - n ) / W ( n )= exp [ ( 4 m / L )J f d s ] .

<

(2) We also see from Eq. (6.1), that a change in sign of the modulating function, from f to -f , gives a similar effect as that of n to -n, i.e., from “left-handed” to “right-handed”. Since the role of the modulating function f is that of simulating the data contained in the lineal structure of the polymer, it appears that the choice of the sign of f is one way of incorporating the observation that the “handedness” of

9

the constituent monomers manifests in the chirality of the polymer. (3) When V = 0, chirality has no effect on the entanglement probabilities. This holds true for the generic case where chirality of the polymer has no influence on the way the macromolecule entangles.

From these, one sees that the model embodies certain features of the “handedness” of a polymer which may lead us to a better understanding of chirality and its influence on the three-dimensional structure of macromolecules.

7 Examples of a Modulating Function An advantage of this model is that the potential, V = f ( s ) 6, where 0 5 s 5 L with L the length of the polymer, can be chosen to simulate the data contained in the one-dimensional structure of the entangled polymer. In particular, one can choose the form of the modulating function f ( s ) . We, therefore, have a framework where an effective potential, chosen to represent the effect of the sequence of monomers making up the macromolecule, can be used in order to determine the over-all structure in three-dimensions such that it departs from the generic case. In short, we have as our input the f (s),and our output the winding probabilities W ( n ) .We now consider two cases below.

7.1

f(s) = k cos (VS)

This choice leads to the potential, V = -kv sin (us) 6, where we could take u as the frequency of the repeating unit in the polymer. From Eq. (5.7) the winding probability becomes,

For a very long macromolecule, L proaches unity and we have,

=

Nl >> 1, the denominator ap-

1

)

2n-nk . k2 sin2 (vL) sin (vL)-

R 47r -47r2n2 R2 w ( n ) zT G e x P ( N12

4NR2v2

(7.2) . , As can be seen in Eq. (7.2),the second exponential modifies the entanglement probabilities for the generic case. The effect of the second exponential may also be viewed by using the expansion, exp ).( = xn/n!. 10

‘

Note, however, that when the frequency has the value v = nn/L, where n = 0, f l , f 2 , ..., Eq. (7.2) reduces to the generic case.

7.2 f(s) = ksp For this choice of the modulating function we can take k to be a positive constant and the possible values of p to be, p = f l ,f 2 , f 3 , ... . When p = -1, -2, -3, ..., the integral f (s) ds in Eq. (5.7) becomes infinite thus damping out the exponential. This implies that W ( n ) = 0, for k > 0. Physically, this would correspond to a stretched or uncoiled polymer. On the other hand, for p = +l,+2,+3, ..., we get, f (s) ds = IcLpfl/ ( p l), and we have from Eq. (5.7),

s

+

(2sn.+ mlkLp+l )2]

R

4n

1

For a very long macromolecule, L

(7.3) 0 3 (4%pp=:])

= N1

>> 1, this becomes,

The second exponential which modifies the generic case inhibits, in general, the coiling of the polymer. Theseobservations imply that, f(s) = ksP, (Ic > 0, p = f l ,f 2 , *3, ...), belongs to a class of modulating functions that can inhibit the coiling of polymers where, V = f (s) 19, becomes a stretching potential.

8

Conclusion

In this paper, we employed the white noise path integral to investigate the morphology of macromolecules. Starting from the entanglement scenario originally studied by Edwards [2] and Prager and Fkisch [3], we simulate the one-dimensional information embodied by an entangled polymer by introducing the potential, V = f(s) 19. We showed that different choices of the modulating function f (s) give rise to different entanglement probabilities. Ideally, one should choose a modulating function f (s) that would best simulate the biochemical data contained in the one-dimensional structure of the entangled polymer to predict the coiling behavior of polymers. The model studied also makes explicit the effect of chirality on winding probabilities. The present study should

11

then lead to more detailed models which could provide additional insights on the study of protein folding and the role of chirality in the globular structure and morphology of macromolecules. Acknowledgement The authors would like to thank L. Streit and F. W. Wiegel for their helpful comments.

References [l]See, e.g., P. Ball, Designing the Molecular World (Princeton Univ.

Press, Princeton, 1994). [2] S. F. Edwards, Proc. Phys. SOC.London 91 (1967) 513-519. [3] S. Prager and H. L. F’risch, J. Chem. Phys. 46 (1967) 1475. [4] C. C. Bernido and M. V. Carpio-Bernido, J. Phys. A: Math. Gen. 36 (2003) 4247-4257. [5] L. Streit and T. Hida, Stoch. Proc. Appl. 16 (1983) 55-69. [6] F. W. Wiegel, J. Chem. Phys. 67 (1977) 469-472 [7] T. Hida, H. H. Kuo, J. Potthoff and L. Streit, White Noise. An Infinite Dimensional Calculus (Kluwer, Dordrecht , 1993). [8] N. Obata, White Noise Calculus and Foclc Space, Lecture Notes in Mathematics, Vol. 1577 (Springer, Berlin, 1994). [9] H. H. Kuo, White Noise Distribution Theory (CRC, Boca Raton, FL, 1996). [lo] See, also, C. C. Bernido and M. V. Carpio-Bernido, J. Math. Phys. 43 (2002) 1728-1736. [ll]I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic Press, San Diego, 1994) p. 927.

12

White Noise Analysis, Quantum Field Theory, and Topology Atle Hahn Institut fur Angewandte Mathematik der Universitat Bonn WegelerstraBe 6, 53115 Bonn, Germany E-Mail: hahnQuni-bonn.de

1

Introduction

Topological quantum field theories provide some of the most interesting examples for the usefulness of path integrals. One of the best known of these examples was discovered in [32] where one particular topological quantum field theory, Chern-Simons theory, was studied and the so-called “Wilson loop observables” (WLOs) were computed explicitly. These WLOs are heuristic path integrals and the interesting thing about the expressions obtained in [32] is that they involve highly non-trivial link invariants like the Jones polynomial, the HOMFLY polynomial, and the Kauffman polynomial, cf. [26, 161. A more thorough study of the WLOs by other methods [18, 8, 5, 171 later led to a breakthrough in knot theory, the discovery of the universal Vassiliev invariant [27]. Unfortunately, it has not yet been possible to establish the aforementioned connection between path integrals and knot polynomials at a rigorous level. In the special case, however, where the base manifold M of the Chern-Simons model considered is of product form the situation looks much more promising and as we will show in the present paper it is reasonable to expect that, at least for some of these special manifolds M, it will eventually be possible 1. to obtain a rigorous definition of the WLOs in terms of Hida distributions (Task 1) and 2. to prove that the values of the rigorously defined WLOs are indeed given by the explicit formulae in 1321 (Task 2).

The present paper is organized as follows. In Sec. 2 we briefly describe what Chern-Simons models are and why, on a heuristic level, they give rise to link invariants. In Sec. 3 we will summarize some recent results for the manifold M = W3 R2 x R, for which Task 1has already been carried out successfully. This manifold 13

has the drawback of being noncompact and for this reason one cannot he sure that the values of the WLOs are given by the formulae in 1321. Fortunately, there is at least one compact manifold for which Task 1 can also he carried out, namely the manifold M = S2 x S1. In Sec. 4 we will give an overview over the results obtained so far for this manifold and we will sketch what remains to he done in order to complete Tasks 1 and 2.

2 2.1

Chern-Simons models, heuristic path integrals, and topological invariants Chern-Simons models

A (pure) Chern-Simons model is a gauge theory on a 3-dimensional connected oriented manifold M without (!) Riemannian metric. The manifold M is often assumed to be compact but sometimes also the noncompact case is studied. The structure group G of the Chern-Simons gauge theory is usually assumed to be compact and connected. Without loss of generality we will assume in the sequel that G is a (closed) subgroup of the group U ( N )with fixed N 2 1. Once the base manifold M and the group G of the model are known the only free parameter of the theory is the so-called “charge” k E B\{O}, which in the case of compact M is assumed to be an integer. We set X := and call X the “coupling constant” of the model. The Lie algebra of G, which we will identify with the obvious Lie subalgebra of u ( N ) ,will be denoted by g. Using this identification we obtain a real (!) bilinear form (., : g x g 3 (A, B ) H - Tr(A . B ) E B on g, which can be shown to he a scalar product. The space of gauge fields, i.e. the space of all g-valued smooth 1-forms on M , will he denoted hy A. What makes Chern-Simons theory special is the fact that the action functional SCS : A -+ C of a Chern-Simons model does not involve a Riemannian metric. It is given by

As M was assumed to be oriented and 3-dimensional, the integral on the righthand side is well-defined even though no measure is involved.

2.2

Heuristic path integrals and topological invariants

From the definition of SCSit is obvious that SCSis invariant under (orientationpreserving) diffeomorphisms. Thus, at a heuristic level, we can expect that the heuristic integral (the “partition function”)

Z ( M ) :=

s

exp(iScs(A))DA

is a topological invariant of the 3-manifold M . With DA above we mean the informal “Lebesgue measure” on the space A. 14

Similarly, we can expect that the mapping which maps every sufficiently “regular” link L = ( 1 1 ~ 1 2 ,... , I n ) in M to the heuristic integral (the “Wilson loop observable” associated to L )

is a link invariant. Here we have used the standard physicists notation P exp for the holonomy of A around the loop li.

2.3

(hiA )

A “Paradox”

For Abelian G the path integral expressions for the WLOs are well understood, see, e.g., [2, 301. For Non-Abelian G the WLOs were evaluated explicitly for the first time in [32]. The explicit expressions obtained in [32] contain the famous Jones polynomial and its two-variable generalizations, the so-called HOMFLY and Kauffman polynomials. These polynomials are highly non-trivial link invariants which were discovered only a few years before [32] was written. Later, a more thorough study of the WLOs by other methods (see Subsec. 2.4 below) led to a breakthrough in knot theory, the discovery of the universal Vassiliev invariant in [27]. Thus we are in a somewhat “paradoxical” situation. On the one hand the heuristic integral expressions (2.1) contain some very deep mathematics. On the other hand it is absolutely not clear whether it is possible to give a rigorous mathematical meaning to these heuristic expressions. As we will try to demonstrate in the sequel things look better if one considers only special base manifolds M and introduces a suitable gauge fixing procedure. Before we do this in Secs. 3 and 4 let us briefly summarize the approaches that have been developed so far for the computation of the WLOs.

2.4

The computation of the WLOs: A short overview

To our knowledge the following approaches have been worked out for the computation of the WLOs for general G. i) Firstly, there is the original approach in [32] which uses arguments from Conformal Field Theory and surgery operations on the base manifold. This approach is the most elegant one. It is non-perturbative, does not involve a gauge fixing procedure and works for arbitrary base manifolds M . The drawback of this approach is that it is not very “explicit”. ii) Another approach is the approach developed in [17] (see also [28]) for the special manifold M = R3. This approach is based on light-cone gauge fixing (which is equivalent to axial gauge fixing) and an additional “complexification” of the coordinates of the points in M . It is non-perturbative and rather explicit. The drawbacks of this approach are, firstly, that it does not work for general M and, secondly, that it leads to the correct values of the WLOs only when certain “ad hoc” correction factors are 15

introduced. Until today there seems to be no convincing explanation why these correction factors have to be used. iii) There is also an approach for the special manifold M = P3which is based on axial gauge fixing without “complexification” of the coordinates of the points in M . Until 1997 this approach was much less developed than the other three approaches listed here and only preliminary results were available, cf. [15], [29], [13]. Then, in [3] it was suggested that white noise analysis should be used for a successful treatment of the axial gauge setting. Finally, in [21, 22, 231 it was shown that the suggestion in [3] is indeed correct and that with the help of white noise analysis the WLOs can indeed be defined and computed rigorously in the axial gauge setting. The advantages of the the axial gauge approach (without complexification) are that this approach is simple and rigorous. The essential drawback is that the corrections which have to be made if one wants to obtain the correct values of WLOs are even more serious than in the “complexified” approach by [ 171, cf. [23]. iv) Finally there is the “Lorentz gauge fixing” approach initiated in 118, 8) and later elaborated by [7, 6, 5, 12, 41. This approach works for arbitrary M and is rather explicit. The drawback of this approach is that it is perturbative and rather complicated.

2.5

Open Questions

When comparing the four approaches listed above a couple of questions arise rather naturally:

- Why does the application of the “non-covariant” gauges, i.e. light-cone and axial gauge (with or without complexification) only lead to the correct values of the WLOs when certain “ad hoc” corrections are carried out later? Are the deviations from the correct values of the WLOs to be seen as defects of these non-covariant gauges or is there another explanation? - Is there an approach which is simple, explicit, and correct? Is there an

approach with these three properties which is even rigorous? If so, this does not necessarily mean that the “paradox” of Subsec. 2.4 is solved’ but it would certainly be an important step towards the resolution of it. It is mainly these two questions that will concern us during the rest of this paper. lNote that by applying a gauge fixing procedure one modifies the heuristic path integral expressions (2.1) so even if one succeeds in making sense of the modified path integral expressions this does not mean that one has also made sense of the original path integral expressions.

16

White noise analysis applied to Chern-Simons models on M = R3 in axial gauge

3 3.1

The basic idea

Let dxi, i = 0,1,2, denote the standard 1-forms on R3. For every A E A we will denote the coordinates w.r.t. (dxi)i=0,1,2 by (Ai)i=0,1,2. We will call A E A axial iff A2 = 0 and we will set A”” := {A E A I A2 = 0 ) . The space A”” has two nice features: 1. The so-called “Faddeev-Popov determinant” AFadpop which is associated to axial gauge fixing is a constant. By “definition” AFadPop is the unique function on A”“ with the property

for every gauge-invariant function x : A --t C. Here DA”” is the (informal) “Lebesgue measure” on A“”. As AFadpop(Aax)is constant we obtain for the special case x(A) = nin(’Pexp(Sli A)) exp(iScs(A)) where L = (II, . . . ,In) is a fixed link in R3

N

1

n T r ( P e x p ( 1 A”“)) exp(iScs(Aa“))DAax.

A”” i

Here

N

1;

denotes “equality up to a multiplicative constant”.

2. For every A““ E A”” we have A”” A A”” A A”“ = 0. Thus we obtain

Taken together these two features “imply” that the heuristic “measure” dpFs(A””) :=

% exp(iScs(Aa”))DAax h

where Z a X ( M ):= exp(iScs(A”“))DA”” is of “Gaussian type” with “covariance operator”

c:=2 T i X .

(-oa, 2) 17

-1

.

3.2

Step 1: Making sense of J’. - .dpFs

If one can make sense of the operator C above as a continuous operator N ---t := Sgeg(W3) 2 A”” then < . >Fs:= s . . . d p F s can be defined rigorously as an element of (N)”corresponding to the Gelfand triple N c H c N’ where H := Lie,(W3,dz). We make the following Ansatz:

N’ where N

where 8;’ : Sg(W3)+ Cr(W3,g)is a left-inverse of the operator 8 2 , i.e. fulfills 8;’ = idsg(p). It is not difficult to see that every such left-inverse 8;’ must be a linear combination of the form s (1 - s) . s E W,where

a;’,

+

1“’ -1

f(zo,zl,t)dt for f E Sg(R3)

(&‘f)(z) =

--oo

05

( % ‘ f ) ( z )=

f ( z o , z l , W for f E S,(R3).

”a

Each operator C, s E

W,which is obtained from Eq.

(3.1) above when inter-

(-oa,

in the way just described is indeed a conpreting the operator tinuous operator N -+ W . Moreover, one can show that although C, depends on the choice of the parameter s, the quadratic form Qax(j) :=>H, j E N c H,does not. As n/ 3 j H exp(-+Qa”(j)) E C is clearly a U-functional the Characterization Theorem allows us to define < . >Fs:= s . . . d p F s rigorously as the unique element of (N)*such that 1

< exp(i(.,j)) >Fs= exp(--Q””(j)) 2

for all j

EN

where on the left-hand side (., .) denotes the canonical pairing n/* x

3.3

N

+

W.

Step 2: Making sense of < n , T r ( P e x p ( L i ( - ) ) )>Fs

After having succeeded in making sense of the heuristic integral functional j” dpFs we can now ask whether it is also possible to make sense of the whole expression < n i T r ( P e x p ( h , ( . ) ) ) >Fs. This can indeed be achieved with the help of two regularization procedures, “loop smearing” and “framing”. These two regularization procedures are described in detail in [21, 22, 231. Here we will content ourselves with a very brief sketch of how they are used:

ni

Loop smearing: We regularize Tr (Pexp(J, (.))) by using “smeared loops” 1: (later we eliminate the variable E by letting E + 0.) One can show that for fixed E > 0 the function n,Tr(Pexp(Jl:(.))) is in the domain ( N )of < . >Fs. Framing: In order to implement the “framing procedure” we first fix a family of diffeomorphisms of W3 with certain properties. In particular, we

($s)s>o

18

demand that ds + idas as s + 0 in a very weak sense. Each diffeomorphism q5s gives rise to a “deformed” version < . >gof < . >E% (later, we let s + 0). Of course, one has to prove that the double limit

W L O ( L ; ~:= ) lim lim < n n ( p e x p ( s-+o € 4 0

i

really exists. This is part of the next step.

3.4

Step 3: Existence proof and computation of W L O ( L ;4 )

For simplicity we will consider only two special cases, namely G = U(1) and G = S U ( N ) . Let us start with the case G = U(1). Before we state the corresponding theorem let us briefly recall the definition of the linking number LK(1,l’) of two given loops 1 , I’ which do not intersect each other. LK(1,l’) is given by LK(1,l‘) := f €(p)

c

PECT(l,l‘)

where cr(1,l’) is the set of all mutual crossings of the planar loops which are obtained by (orthogonally) projecting 1 and 1’ to the xo-xl-plane. To each crossing p E cr(1,l’) a “sign” ~ ( p E) {-1, l} is associated according to the following pictures:

Figure 1:

Figure 2:

~ ( p=) -1

~ ( p=) 1

Theorem 1 Let G = U(1). Then for every “admissible” link L and every “admissible” framing q5 := ( q 5 s ) s > ~ the double limit

exists and we have

where lkj = lims+0 L K ( l j ,d5 o l j ) .

19

For an exact definition of the notion “admissible” for links and framings see 121, 221. The “wraith” w ( L ) of a link L = ( 1 1 ~ 1 2 , .. . , I n ) is given by

w ( L ) :=

c

4P)

PEWL)

where V(L) is the set of all (mutual and self) crossings of the planar loops which are obtained by (orthogonally) projecting the loops Z1, 12, ..., 1, to the $0-xl-plane. Two loops I , 1’ which are very ‘Lclose”to each other but do not intersect can be considered to be the boundary of a “ribbon” obtained by interpolating l ( t ) and l’(t) for every t E [0,1]. The number of “twists” of this ribbon will be denoted by twist(1,l’).

Theorem 2 Let G = S U ( N ) . Then for every “strongly admissible’’ link L and every “strongly admissible” framing 4 := ( q 5 s ) s > ~ the double limit

exists. wLO(L; 4) is independent of the “loop smearing axis” which was fixed in the course of the loop smearing procedure af and only i f X E 22. I n this case we have WLO(L; 4)= N # ~ exp(-+ t i ) exp(-+ w(L)). (34

jsn Here #L is the number of components of L and

t j := lirn,-o

twist(lj, 4s0 l j ) .

For an exact definition of the “loop smearing axis” and the notion “strongly admissible” for links and framings, see [23].

3.5

Comparison with the results obtained by the other approaches

According to the standard literature, cf., e.g., [18], [17],we should have WLO(L; 4)= HOMFLYL(exp(XrzN), 2isin(Xr)) x x

fl expO\ri

+tj)

NX1

exp(X7ri -w(L)) (3.3)

i5n

for every X E A := {t*,&&, ...}. Here HOMFLYL is the HOMFLY polynomial which is associated to the link L . The set A consists of those values of X for which the charge k = is an integer and for which equation (3.3) gives rise to values for the Wilson loop observables which are compatible with the “unitarity” of the theory (cf. pp. 168f in [17]).

20

In order to compare equation (3.3) with (3.2) let us introduce the function fL : R\Z 3 X

H

HOMFLYL(exp(XniN), 2isin(Xn)) x

EC

fL is a well-defined function on R\Z which can be extended uniquely to a continuous function f~ on all of R. It is easy to see that ~ L ( X ) = N # L if X E 2 2 . Clearly, for X E 2 2 we have e x p ( - F ) = exp(Xnz(*)). Thus equation (3.2) can be considered to be the “special case” of equation (3.3) for X E 225. Theorem 2 raises the question whether one should perhaps replace the set A by 2 2 . For all X E 2 2 the theory should again be unitary. Of course, the charge k = l will then not be an integer, but we doubt whether it makes sense to demand k E 2 if the base manifold M of the Chern-Simons model considered is noncompact like in the case M = R3.This leads us to the following conjecture.

Conjecture. The problems that appear when applying light-cone gauge and axial gauge fixing to Chern-Simons models on R3 have nothing to do with these gauges but with the non-compactness of R3. Fortunately, there is a good chance of finding out wether this conjecture is true or not. This is because there is at least one compact manifold, namely M = S2 x S’, for which a gauge fixing is available that is very similar to axial gauge in the case of It3. This gauge was called “Torus gauge” in [9]. In Sec. 4 we will show how, using torus gauge fixing, one can find a rigorous representation of the WLOs in terms of Hida distributions also for Chern-Simons models on S2 x S’. We expect that by computing the WLOs explicitly one will get an answer to the question whether the conjecture above is true or not.

White noise analysis applied to Chern-Simons models on A4 = S2 x S1 in torus gauge

4 4.1

Torus Gauge

In order to make the similarities between axial gauge and torus gauge (to be defined below) more explicit let us first consider the manifolds of the form M = C x R. Let t denote the global coordinate M R given by t(u,s) = s, u E C, s E R. The global coordinate t gives rise to a 1-form d t on M . By lifting the constant vector field on R taking only the value 1 to the manifold M = C x R with the help o f t we also obtain a vector field on M . This vector field will be denoted by Clearly, we have d t ( & ) = 1. Let us now introduce the subspace A l A’ = { A E dl A ( & ) = 0 ) --f

&.

of

A. Clearly, every A E A can be written uniquely A

= A’

21

+ Atdt

in the form

with A’ E A’ and At E C“(M, g). Note that in the special case where C = R2 and consequently M = R2 x R 2 R3 the space A’ coincides with A”” defined in Sec. 3. In this case the following three statements are clearly equivalent: A is axial, A = A’ and At = 0. After these preparations let us now consider manifolds M of the form M = C x S’. Even though the mapping t : C x S’ --t S1 with t ( o , s ) = s is not a global coordinate it can be used to “lift” the standard 1-form dt and the standard vector field on 5’’to a 1-form resp. vector field on C x S’. The As lifted 1-form resp. vector field on M will again be denoted by dt resp. before we can now introduce the space

&

4.

A’ := { A E A I A ( & ) = O}. Again every A E A can be written uniquely in the form

A

= A’

+ Atdt

with A’ E A’ and At E C m ( M , g ) . However, there is a crucial difference between the case M = C x R and the case M = C x S’. For M = C x R the condition At = 0 “defines” a gauge. More precisely: Every 1-form A E A is gauge equivalent to a 1-form in A’- = { A E A I At = 0 ) . By contrast for M = C x 5’’the condition At = 0 does not define a gauge. There are 1-forms which are not gauge equivalent t o any 1-form in A’. For example this is the case for any 1-form A for which the holonomy Pexp(Jco A ) around the loop I,, : S’ 3 s H ( 0 0 , s) E M where oo E C is a fixed point is not equal to 1. This follows immediately from the two observations that, firstly, the holonomies are invariant under gauge transformations and, secondly, we clearly have P exp( A’) = 1 for every A’ E A’. 1-0 Thus, in order t o obtain a proper gauge we have to weaken the condition At = 0. There are two natural candidates for such a weakened condition. 1. Option: Instead of demanding At(o,s) = 0 for o,t we just demand that At(o,s) is independent of the second variable s, i.e. we demand that At = B holds where B E C“ ( C ,g) (“Quasi-axial gauge”) 2. Option (better): We demand, firstly, that At(a,s) is independent of the second variable and, secondly, that it takes values in the Lie algebra t of a fixed maximal torus T c G (“Torus gauge”),

Thus we arrive a t the following definition

Definition 1 Let T be a maximal torus of G. A 1-form A E A is said to be “in the T-torus gauge” iff there is a A’ E A’ and a B E C”(C,t) such that

A

= A’

+ Bdt holds.

In the next subsections we will restrict ourselves to the special case C = S 2 .

22

4.2

The Faddeev-Popov-Determinant of Torus gauge fixing

From now on we will set C := S2. The aim of this subsection is to identify the Faddeev-Popov-Determinant a F a d p o p which is associate to torus gauge fixing. By “definition” A F a d p o p is the unique mapping on A‘ x Cm(s2,t) such that

(4.1) holds for every gauge invariant function x : A 4 C. Here DA’ denotes the (informal) “Lebesgue measure” on A’ and D B the (informal) “Lebesgue measure” on Cm(S2,t). It is possible to compute A F a d p o p explicitly and if one does so one obtains the following heuristic equation:

+ Bdt) = 1 det(& + ad(B)) I det(lgu

AFadPop(A’

-

exP(ad(B))lgu)

Here go denotes the (., .),-orthogonal complement of t in g. The special case of Eq. (4.1) in which we are interested is the case obtained by taking x ( A )= Tr(Pexp(h, A ) ) exp(iScs(A)) where L = (11,. . . ,In) is a fixed link in S2 x S1. In this special situation we have

ni

4.3 A Formula for Scs(A‘-

+ Bdt)

Let us identify the space A’ with the space C”(S1,dsz) of “smooth”2 ds2valued mappings on S1 in the obvious way. Here As2 denotes the space of g-valued 1-forms on S2.Moreover, we introduce the bilinear form

< ., . >sz: dsz One can show that for all A ’ Scs(A’

x Asz 3 E

(0, 01’)

H

A’- and all B

L E

n(0 A 1 01’)

EC

Cm(S2,t) we have

+ Bdt)

=-&ll[
2For a definition of the notion “smooth” here, see [24]

23

1

>sz -2sz dt

(4.3)

Consequently, for fixed B , the mapping

dl

3 A'

Scs(A'

+ Bdt) E C

is quadratic. This point will be of crucial importance in the sequel.

4.4

Introduction of a scalar product

Let us now fix an auxiliary Riemannian metric g on S 2 and let pg denote the Riemannian volume measure on S2 which is induced by g. Obviously, the bilinear form

is a scalar product on A'. Here (.,.), denotes the fibre metric on the bundle Hom(TC, g) E TC' @ gwhich is induced by the metric g and the scalar product (.,.)* on g. The Hodge star operator * : dsz 4 dsz induces a linear automorphism of A' C" (S', As*)which will also be denoted by * and which is explicitly given by (*A')@) = *(AL(t)) V t E S1 With the help of << ., . >> and * we can now rewrite Eq. (4.3) in the form Scs(A'+Bdt)

=

-&

<< A', (*o(&+ad(B))).A' >> -2 << A',*.dB

1

>>

(4.4)

It is tempting to conclude from (4.4) that Scs(A' =-

+ Bdt)

& << (A'-

(&+ad(B))-l.dB),

(*o(

&+ad(B))). (A'

-(&+ad(B))-l .dB) >> (4.5)

If one could make sense of the latter equation, one could conclude at an informal level that the (informal) 'Lmeasure"exp(iScs(A'+ Bdt))DAL is of "Gaussian type" with "mean"

(2+ ad(B))-l -y(*o

4.5

. d B and a

"covariance operator"

(& +ad(B)))-'

The decomposition dl = d' @

+

1

However, there are two problems with Eq. (4.5): Firstly, (& ad(B)))ad(B) is not injective, and secondly, the operator does not exist because 0 ad(B)) is not symmetric w.r.t. the scalar product << ., . >>.

* (g+

&+

24

Both problems can be solved by introducing the decomposition A' = d' @

d$ where d' d:

:= {A'

I rdSZ,t(A'(tO))= 0) I Vt E S' : A'(t) = A'(t0)

A' := {A' E A' E

E dsz,t}

where t o is a fixed point in S1 and 7Tds2,1 : dsz 2 dSz,go@ dsz,t3 dSz,t the canonical projection. Here dSz,gO(resp. dsz,t)denotes the space of smooth go-valued (resp. t-valued) 1-forms on C. It can be shown that the restriction (& ad(B)),al of the operator (&

+

+

(6+

ad(B)) onto d' is injective and that the operator * o ad(B)) IAI is symmetric w.r.t. the scalar product << ., . >. Finally, by extending (& ad(B)) in a suitable way to the space

A'

:= d' @

+

{A: . (iit(.)- 1/2) I A: E dc,t}

+

where is1 : [0,1) 3 t H exp(2ri(to t ) ) E S' c C one can achieve that the extended operator, which will also be denoted by (& ad(B)) , is a bijection A' + A'. Thus m ( B ):= (& ad(B))-' . dB (4.6)

+

+

is a well-defined element of

Scs(A' Scs(A'

A'.

It is not difficult to show that

+ A: + Bdt) = Scs(A' + Bdt) + & < A:, dB >sz, + Bdt) = --4Kk <<'A - m ( B ) ,(*a (& + ad(B))) . (A'

(4.7) - m ( B ) )>>

(4.8) holds, which means that the heuristic integral functional

1..

.exp(iScs(A'+

B~~))DA'

is of "Gaussian type" with "mean" m ( B ) ,"covariance operator"

C ( B )= - Y ( * o ( - & + a d ( B ) ) ) - ' and "mass" I d e t ( g +ad(B))/-'/' of B).

4.6

(4.9)

(up to a multiplicative constant, independent

A preliminary heuristic formula for WLO(L)

Combining Eqs. (4.2), (4.7), (4.8) we arrive at

WLO(L)

25

&

There is a curious thing about this equation: In the expression I det( +ad(B)) I which appears above the operator + ad(B) denotes the obvious operator P ( S 2 x S',g) C"(S2 x S',g). Now, at a heuristic level, the determinant of this operator should equal3 the root of the determinant of the operator

&

--f

(& +ad(@)

: Cw(S1,dSZ) 4

C"(S1,dsz).

Thus, heuristically, the "measure"

d f i i ( A ' ) := 1 det(&

+ ad(B))I exp(iScs(A' + Z3dt))DAl

has mass 1 and we can rewrite Eq. (4.10) as

x det(l,, - exp(ad(B))l,,) exp(i&

< Ag,dB >sz)DA:

@ DB.

(4.11)

4.7 The final heuristic formula for WLO(L) So far we have neglected one topological subtlety. Above we claimed that torus gauge fixing is a proper gauge k i n g . However, strictly speaking, this is only true if the oriented surface C in Subsec. 4.1 is non-compact. If the surface C is compact like in the case C = S2 we can not expect Eq. (4.11) to hold without modification. The modification that takes care of the topological subtleties over the set which we have just mentioned involves a summation CnEISZ,G,Tl [S', G / T ]of free homotopy classes of mappings from S 2 to G/T and, for each n E [S2,G / T ] ,n # 0, the inclusion of a 1-form A$,,,(n) with a singularity in a fixed point 00 of C. More precisely, the modification of Eq. (4.11) is given by

WLO(L)

exp(i& < A:,dB >p)DA: 8 DB. (4.12)

4.8 The Program The heuristic equation (4.12) can be used as the starting point for the search of a rigorous definition of the WLOs in terms of Hida distributions. In order to make rigorous sense of the right-hand side of Eq. (4.12) one can proceed in 5 steps: 3 ~ f .19, 101; note that we consider the special cme C = S2 where the Euler characteristic x ( C ) equals 2

26

0

Step 1: Make rigorous sense of the heuristic integral functional

0

Step 2: Make rigorous sense of the whole expression SJ.

A: 0

s.. d f i i ( a i ) *

niTr(Pexp(h, A*I+

+ Ak,,(n) + B d t ) ) d f i i ( A l )

Step 3: Make rigorous sense of the heuristic integral functional

J

. . . e x p ( i 3 < A t , dB >Sz)DA,I 8 DB

d$xCw(S2,t)

as a Hida distribution of “Gaussian type”. 0

Step 4: Make rigorous sense of the total expression on the right-hand side of Eq. (4.12) Step 5: Compute the expression in Step 4 explicitly.

In [24, 191 we have already completed Steps 1-3 for arbitrary G using “loop smearing” and “framing” in a similar way as in Sec. 3 and, additionally, Steps 4 and 5 in the special case where G is Abelian. We plan to complete the last two steps also for Non-Abelian G in the near future, cf. [20].

4.9

Some Details for Step 1

Let 7 - 1 ~denote the Hilbert space L2-r(Hom(TC,g),pg) of L2-sections of the bundle Hom(TC,g) w.r.t. the measure pg and let H denote the Hilbert space L$& (S’, dt) of square-integrable HE-valued functions on S’. Moreover, let us identify the spaces AL and d l with the obvious subspaces of 7-1. Then the operator C ( B ) : A l --* d l can be considered as a densely defined bounded symmetric operator on 7-t = L&=(S’,dt). Setting N := A’ we obtain a Gelfand triple N c ‘H c N*. The informal integral functional . . .d f i i can now be defined rigorously as the unique element of (N)*such that

@A

s

@i(exp(z(.,j)))= exp(i << j , m ( B ) >>x)exp(-i << j , C ( B ) j >>x) (4.13) holds for all j E N with m ( B ) and C ( B ) given as in (4.6), (4.9). Here (., .) : -+ W denotes the canonical pairing and << ., . >>xthe scalar product of ‘H.

N* x N 5

Conclusions and Outlook

In this paper we have explained how white noise analysis can be applied successfully to the study of Chern-Simons models on W3 W2 x W and S2 x S1. If it turns out after the completion of the 5 steps in Subsec. 4.8 that the torus gauge fixing approach of Sec. 4 applied to Non-Abelian G will lead to the correct knot polynomial expressions for the WLOs this will clarify most of the open questions listed in Subsec. 2.5. In particular, it would provide a strong argument

27

in favor of the conjecture made in Subsec. 3.5 and it would demonstrate that it is indeed possible to establish the relations discovered in [32] between the knot polynomials and the heuristic path integral expressions for the WLOs at a mathematically rigorous level.

Acknowledgements: I would like to express my gratitude to Prof. Dr. T. Hida for giving me the opportunity to contribute to the very stimulating Conference in Nagoya last November and to the Proceedings.

References [l] S. Albeverio, A. Hahn, and A. N. Sengupta. Chern-Simons theory, Hida distributions and state models. Inf. Dim. Anal. Quant. Prob., 6, Supplementary volume 1:65-81, 2003. [2] S. Albeverio and J. Schafer. Abelian Chern-Simons theory and linking numbers via oscillatory integrals. J . Math. Phys., 36(5):2135-2169, 1994.

[3] S. Albeverio and A.N. Sengupta. A Mathematical Construction of the Non-Abelian Chern-Simons F'unctional Integral. Commun. Math. Phys., 186:563-579,1997. [4] D. Altschuler and L. Freidel. Vassiliev knot invariants and Chern-Simons perturbation theory to all orders. Comm. Math. Phys., 187:261-287, 1997.

[5] S. Axelrod and I.M. Singer. Chern-Simons perturbation theory. In Catto, Sultan et al., editor, Differential geometric methods in theoretical physics. Proceedings of the 20th international conference, June 3-7, 1991, New York City, NY, USA, volume 1-2, pages 3-45. World Scientific, Singapore, 1992. [6] S. Axelrod and I.M. Singer. Chern-Simons perturbation theory. 11. J. Differ. Geom., 39(1):173-213, 1994. [7] D. Bar-Natan. On the Vassiliev knot invariants. Topology, 34:423-472, 1995. [8] D. Bar-Natan. Perturbative Chern-Simons theory. J . Knot Theory and its Ramijcations, 4:503-547, 1995. [9] M. Blau and G. Thompson. Derivation of the Verlinde Formula from ChernSimons Theory and the G/G model. Nucl. Phys., B408(1):345-390,1993.

[lo] M. Blau and G. Thompson. Lectures on 2d Gauge Theories: Topological Aspects and Path Integral Techniques. In E. Gava et al., editor, Proceedings of the 1993 Weste Summer School on High Energy Physics and Cosmology, pages 175-244. World Scientific, Singapore, 1994. 1111 M. Blau and G. Thompson. On Diagonalization in Mup(M, G). Commun. Math. Phys., 171:639-660, 1995. 28

[12] R. Bott and Taubes C. On the self-linking of knots. J. Math. Phys., 35( 10):5247-5287,1994. [13] A. Cattaneo, P. Cotta-Ramusino, J. Frohlich, and M. Martellini. Topological B F theories in 3 and 4 dimensions. J. Math. Phys., 36(11):6137-6160, 1995. [14] M. de Faria, J. Potthoff, and L. Streit. The Feynman integrand as a Hida distribution. J. Math. Phys., 32(8):2123-2127, 1991. [15] Shmuel Elitxur, Gregory Moore, Adam Schwimmer, and Nathan Seiberg. Remarks on the canonical quantization of the Chern-Simons-Witten theory. Nuclear Phys. B, 326(1):108-134, 1989. [16] P. Freyd, J. Hoste, W. Lickorish, K. Millett, A. Ocneau, and D. Yetter. A new polynomial Invariant of Knots and Links. Bulletin of the AMS, 12(2) ~239-246, 1985 [17] J. F'rohlich and C. King. The Chern-Simons Theory and Knot Polynomials. Cornrnun. Math. Phys., 126:167-199, 1989. [18] E. Guadagnini, M. Martellini, and M. Mintchev. Wilson Lines in ChernSimons theory and Link invariants. Nucl. Phys. B, 330:575-607, 1990. [191 A. Hahn. Chern-Simons models on S2 x S', torus gauge fixing, and link invariants 11. In Preparation. [20] A. Hahn. Geometric derivation of the R-Matrices of Jones and lhraev. In Preparation. [21] A. Hahn. Chern-Simons Theory on R3 in axial Gauge. Ph.D. Thesis, Bonner Mathematische Schriften Nr. 345, 2001. [22] A. Hahn. Chern-Simons theory on W3 in axial gauge: a rigorous approach. J. Funct. Anal., 211(2):483-507, 2004. [23] A. Hahn. The Wilson loop observables of Chern-Simons theory on R3 in axial gauge. Cornrnun. Math. Phys., 248(3):467-499, 2004. [24] A. Hahn. Chern-Simons models on S2 x S1, torus gauge fixing, and link invariants I. J. Georn. Phys., 53(3):275-314, 2005. [25] T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit. White Noise. An infinite dimensional Calculus. Dordrecht: Kluwer, 1993. [26] L. Kauffman. Knots. Singapore: World Scientific, 1993 [27] M. Kontsevich. Vassiliev's knot invariants. Adw. in Sow. Math, 16(2):137150, 1993.

29

[28] J.M.F. Labastida. Chern-Simons gauge theory: ten years after. In Trends in theoretical physics 11 (Buenos Aims 1998), pages 1-40. AIP Conf. Proc., Amer. Inst. Physics, Woodbury, New York, 1999. [29] G. Leibbrandt and C. P. Martin. Perturbative Chern-Simons theory in the light-cone gauge. The one-loop vacuum polarization tensor in a gaugeinvariant formalism. Nuclear Phys. B, 377(3):593-621, 1992. 1301 P. Leukert and J. Schafer. A Rigorous Construction of Abelian ChernSimons Path Integrals using White Noise Analysis. Rev. Math. Phys., 8(3) :445-456, 1996. [31] L. Streit and T. Hida. Generalized Brownian functionals and the Feynman integral. Stochastic Process. Appl., 16(1):55-69, 1984. [32] E. Witten. Quantum Field Theory and the Jones Polynomial. Commun. Math. Phys., 121:351-399, 1989.

30

A topic on noncanonical representations of Gaussian processes Dedicated to Professor T. Hida for his 77th birthday

YUJIHIBINO Faculty of Science and Engineering] Saga University 840-8502] Saga, JAPAN Abstract Give a noncanonical representation of a Gaussian process X. If the Hilbert spaces H t ( X ) = L S { X ( s ) ;s 5 t } satisfy a certain condition, then we can form the Brownian motion that is used for the canonical representation of X.

Keywords: Gaussian processes; Canonical representations; Noncanonical representations; Brownian motions.

AMS Subject Classification: 60G15, 60G35

1

Introduction

Let a Gaussian process X = { X ( t ) t; 2 0) be given by a Wiener integral

X ( t )=

l

F ( t ,u)dB(u),

(1)

where B is a Brownian motion and F is a nonrandom function. In the representation (l),Bt(X)(r a { X ( s ) ; s 5 t } ) is smaller than or equal to & ( B ) for each t 2 0. If &(X) = & ( B ) holds, the representation (1) is canonical with respect to the Brownian motion B (see [4]).We remark that &(X) = & ( B ) is equivalent to H t ( X ) = H t ( B ) , where H t ( X ) is a closed linear span of { X ( s ) ;s 5 t } , since X is Gaussian. The concept of canonical representation was originated by Lkvy [4].For the canonical representation, we understand that the information generated by the past of X up to time t is equal to that of B . In this case, the randomness of X is equally undertaken by B , which is called the innovation of X. It is known that the canonical representation is essentially unique, if it exists (see [2, 31). Generally speaking, a centered Gaussian process is determined by its covariance. However, it is difficult to obtain the canonical representation 31

or the multiplicity directly from the given covariance, except for some special cases such as stationary processes and so on. Even if a representation like (1) is given, it may not be canonical. We occasionally met noncanonical representations (see [4],[5], etc.) In the joint paper [l],it has been given how to construct a noncanonical representation of a Brownian motion so as to be independent of a given finite-dimensional subspace. The class includes examples of noncanonical representations given by LBvy. In this article we shall propose a method of constructing the canonical representation when we are given a noncanonical representation of the form ( 1 ) having a finite-dimensional orthogonal complement of H t ( X ) in Ht(B). The fact shows a kind of usefulness of noncanonical representations. The base of H t ( B ) 8 H t ( X ) is obtained by solving the integral equation derived from the Hida criterion [2].

2

Noncanonical representations

It is known in [l]that, for any N E N, we can construct a noncanonical representation of a Brownian motion having a given N-dimensional orthogonal complement as in the following way: Let 91, g 2 , . . . ,g~ E Lfo,[O,m) be linearly independent in L2[0,t ] for any t > 0. Define a Volterra-type integral operator Kg : L2[0,co)+ L2[0,m) bv

(Jig i ( u ) g j (u)du) -'.

where r(t)-' = (rij(t)) = r(t)is invertible for any t linearly independent.

> 0, since g ( t )

It is noted that the matrix

= ( g l ( t ) , g a ( t )., . . , g N ( t ) } is

Theorem 1 ([l, Theorem 2.11) Define a Gaussian process B, b y

Then B, is a Brownian motion and has a noncanonical representation with respect to B : H t ( B ) = H t ( B g ) EBLS where

Kl

{I'

g j ( u ) d B ( u ) ; j= 1 , 2 , . . .

is the formal adjoint operator of K,.

32

We prefer to write (3) symbolically

d,(t) = ( I - K,)B(t),

Bg(0)= 0.

By using the theorem above, the canonical representation of X can be obtained from a noncanonical representation having a finite-dimensional orthogonal complement H t ( B )e H t ( X ) as follows:

Theorem 2 Let a Gaussian process X be given b y (1). If it is a noncanonical representation satisfying

then X has unit multiplicity, and has the canonical representation

X(t)=

l-

F ( t , u)dB,(u)

with respect to B,, where B, is defined b y (3) and F(t,.) = ( I - K,)F(t,.). Proof: The Brownian motion B, defined by (3) satisfies (4). Since ( 5 ) holds, we have Ht(B,) = H t ( X ) for any t > 0. Because of the uniqueness of canonical representation, B, is an innovation of X . Thus the multiplicity of X is one. Suppose the representation (6) of X is canonical. Since t

X(t) =

F(t,U)B,(U)dU JO

= J d ' F ( t , " ) ( I - K,)B(u)du =

I ' ( I - K l ) F ( t ,u)B(u)du,

wesee ( I - K i ) F ( t , . ) = F ( t , . ) . Since (I-K,)(I-K;) F ( t , .) = ( I - K,)F(t, .).

= I , wecanconclude 0

If X defined by (1) is noncanonical with respect to B , clearly H t ( B ) 2 H t ( X ) for some t > 0. When the orthogonal complement H t ( B ) 0 H t ( X ) is finite-dimensional and we can get ( 5 ) , then Theorem 2 can be applied in order to obtain the canonical representation of X.

33

Remark If we could construct, for a system of functions g1,$2, .. . E Lfo,[O,oo), a noncanonical representation of a Brownian motion B having an infinite-dimensional orthogonal complement:

then, even for a given noncanonical representation of X having an infinitedimensional orthogonal complement, the innovation and the canonical representation of X would be obtained by applying the idea of Theorem 2.

Acknowledgments. The author would like to express sincere gratitude to Professor T. Hida for his constant encouragement. Thanks are due to Dr. H. Muraoka for fruitful discussion with him.

References [l] Y . Hibino, M. Hitsuda and H. Muraoka; Construction of noncanonical representations of a Brownian motion. Hiroshima Math. J . 27 (1997), 439-448.

[a] T. Hida; Canonical representations

of Gaussian processes and their applications. Mem. Coll. Sci. Univ. Kyoto 33 (1960), 10S155.

[3] T. Hida and M. Hitsuda; Gaussian Processes, Representation and Applications, Amer. Math. SOC.(1993).

[4]P. L&y; A special problem of Brownian motion and a general theory of .Gaussian random functions. Proc. of 3rd Berkeley Symp. Math. Stat. and Prob. 2 (1956), 133-175.

[5] H. P. McKean, Jr.; Brownian motion with several-dimensional time, Theor. Probability Appl. 8 (1963), 335-354.

34

Integral Representation of Hilbert-Schmidt Operators on Boson Fock Space' U N CIG JI DEPARTMENT OF MATHEMATICS RESEARCH INSTITUTEOF MATHEMATICAL FINANCE CHUNGBUK NATIONAL UNIVERSITY 361-763 KOREA CHEONGJU E-MAIL: uncigj iQcbucc. chungbuk.ac .kr

Abstract

An integral representation of Hilbert-Schmidt operators on Boson Fock space is proved with explicit forms of integrands which is a quantum version of (classical) Clark-Haussmann-Ocone formula.

Mathematics Subject Classifications (2000): Primary: 60H40; Secondary: 81925. Key words: white noise theory, Fock space, integral kernel operator, operator symbol, Hilbert-Schmidt operator, chaotic expansion

1

Introduction

The quantum stochastic calculus of It6 type formulated by Hudson and Parthasarathy [9] has been extensively developed in [20], [27] and the references cited therein. In particular, the stochastic integral representations of quantum martingales have been studied by many authors (see [2], [lo], [12], [19], [20], [28], etc). An integral representation of Hilbert-Schmidt martingales was studied in [ll]. The chaotic expansion of Hilbert-Schmidt operators was established in [3] and more general theory was developed in [14]. On the other hand, the white noise theory initiated by Hida [7] has been considerably developed as an infinite dimensional distribution theory with wide applications: to stochastic analysis, Feynman path integral, quantum physics and so on, see [S], (181 and (221. Recently, white noise approach to quantum stochastic calculus is successively studied in [13], [15], [23][26]. Specially, in [l6], explicit forms of integrands in the integral representation of quantum martingales was obtained by using the quantum white noise derivatives. In this paper, by using the white noise theory we study an integral representation of Hilbert-Schmidt operators on Boson Fock space with explicit forms of integrands. This paper is organized as follows: In Section 2 we recall the basic notions in white noise theory. In Section 3 we review the theory of white noise operators with its chaotic expansions. In Section 4 we study an integral representation of Hilbert-Schmidt operators on Fock space. 'This work was supported by grant (No. R05-2004-000-11346-0) from the Basic Research Program of the Korea Science & Engineering Foundation.

35

Preliminaries

2 2.1

White Noise Triplet

Let H = L2 ( R +, d t)be the (complex) Hilbert space of L2-functions on R, = [O,m) with respect t o the Lebesgue measure d t and the norm of H is denoted by 1 . lo. Let A be a selfadjoint operator (densely defined) in H satisfying that there exist a sequence m

~ < X O ~ X ~ < X Z < . . I.( ,A - ' I I ; , = C X y 2 < m , j=O

and an orthonormal basis { e j } g o of H such that Aej = A j e j . For p E R we define m

I € ;I

I

X?I (€, e j ) 12,

= APE I: =

E

E

H.

j=O

Now, for p 2 0, setting Ep = {< E H ; I E ,1 < 03) and defining E-, to be the completion of H with respect to 1 . I-, we obtain a chain of Hilbert spaces {E, ; p E R}. Define their limit spaces: E = proj lim E,, E* = ind lim E-,, P-m

p-m

where E* is the strong dual space of E. Identifying H with its dual space, we have

E C E,

cH

= L 2 ( R + , d t ) c E-,

c E*,

p 2 0.

(2.1)

(Al) for each function E E E there exists a unique continuous function € ( t )= F(t) for almost all t 2 0;

on R+ such that

From more general study in [22],we assume that

<

(A2) for each t 2 0 the evaluation map 6, : H E(t), E E E , belongs t o E' and the map R+ 3 t H ht E E* is continuous with respect to the strong dual topology of E". Let p E R and r(E,) be the (Boson) Fock space over E,, i.e.,

F(E,)=

EP

{

m

+ = ( f " ) r = O ; f " E E ~ ,II+Il;=~n!lf"I;<.. n=O

1

,

where is the symmetric n-tensor power of E,. By taking Fock space from (2.1), we have the natural inclusions: P 2 0. r(E,) c r ( H ) c r(E-,), By the general duality theory, r(E-,) is the strong dual space of r(E,). Setting r ( E ) = proj limF(E,),

r ( E ) * = indlimr(E-,), ,-a

P-m

we have a complex Gelfand triple:

r ( E )c r ( H ) c r ( E ) *

(2.2)

which is referred to as the Hzda-Kubo-Takenaka space [17], see also [7, 8, 18, 221. The canonical C-bilinear form on r ( E ) * x r ( E ) is denoted by ((., .)).

36

Admissible Triplet

2.2

For p E R we define m

1 1 4 1;

4 = (fn)E r ( ~ ) .

= Cn!e2pnIfnI& n=O

For p

2 0 we put Gp = {4 = (fn)

r ( H ) with respect to (11 . I I - p .

Then

G = proj lim

C

r ( H ) ; 1 1 4 Il p < ca} and G-, to be the completion of E R} form a chain of Hilbert spaces satisfying

E

{Gp ; p

Gp C

=r(H)c

LPc G* = ind lim GPp. P-m

P-m

Note that we have

B is a countable Hilbert space and that G and G+ are mutually dual spaces. Then, r ( E )C

B c r ( H ) c G* c r ( E ) *

of which the proof is immediate from the definition of the norms. The canonical C-bilinear form on 8' x G is denoted by ((., .)) again.

3 White Noise Operators A continuous linear operator from r ( E ) into r ( E ) *is called a white noise operator. The ) with the bounded space of all white noise operators is denoted by L ( r ( E ) , r ( E ) *equipped convergence topology. 3.1 Operator Symbol and Integral Kernel Operators For each

< E E,

,...)

4 < = ( l , +P .'>$

is called an exponential vector or a coherent vector. Note that the exponential vectors { 4 ~ ; E E } span a dense subspace of r ( E ) ,hence of Gp for all p E R and of 9'. Thus each E E L ( r ( E ) r, ( E ) * )is uniquely specified by its matrix elements with respect to the exponential vectors. We put

<

-=(& 11) = (Wt,4 4 h

>

(7

11

E

(3.1)

which is called the symbol of Z. Theorem 3.1 [21, 221 A C-valued function 0 on E x E is the symbol of a white noise operator Z E L ( r ( E ) r, ( E ) * )zf and only i f (i) 0 is Giteaux entire; (ii) there exist C

2 0, K 2 0 and p 2 0 such that

P(E,d I I C ~ X PK ( I E I: + I 7 I:),

37

E,7 E E.

For each given qrn E (E@('+m))*, 1,m 2 0, an integral kernel operator

=- = -

=l,m(qm)

-

= =l,m(Kl,m)

is in L(F(E),r(E)*) satisfying

8(<,q)= (Kl,rn<@m, q@l)e((,q) = (qm, q@1C3<@m)e(E39),<,q E E ,

(3.2)

where K I , E~L(E@",(E@')*) corresponds to tqm E (E@('++"))* under the kernel theorem. The existence of the integral kernel operator is immediate from Theorem 3.1 since Z given in (3.2) satisfies the conditions (i) and (ii) in Theorem 3.1. For each y E E*, the annihilation operator ay and the creation operator a; satisfy h

&(<,d= (Y,<)e(S'9)?

a3y(<,17)=

(Y,17)e(€'",

< , o eE

Then for any ( E E , a< and a; are continuous linear operators acting on r ( E ) (or G). Also, a( has a unique extension to a continuous linear operator acting on r ( E ) *and a; E L ( r ( E )r, ( E ) ) .For each t 6 R+, we write at = and a: = a;*. Note that for each @ E G*, Dt@ = at@ is well-defined for almost all t E R+. Moreover, we have the following

Proposition 3.2 ( [ l ][15]) , For each @ = (F,) E G*, Dt@ = ((n+l)F,+l(t, .))Fz0 for almost all t E R+. Moreover, if @ E Gp and q < p - log 4,then Dt@ E Gq for almost all t 6 R+. Remark 3.3 The kernel distribution of an integral kernel operator El,rn(+,)

is uniquely

determined whenever taken from the subspace

(E@('+m))* sym(l,m) = { K l , r n E (E@('+~))* ; si,rn(Kt,m) = nl,m}

7

where sl,+ is the symmetrizing operator with respect to the first 1 and the second m variables independently. 3.2

Chaotic Expansion of Hilbert-Schmidt Operators

Let p , q E R. For each K I ,E~L(Efm, Eqg')there exists a unique operator &,(Kl,J L ( r ( E p )r,( E q ) )such that

4,m(K1,~)-(<, V ) = ( K I , ~
t,1) E E

E

(3.3)

)I K I ,\lop,~ see [14]. Moreover, K I , E~ L2(Efm,E p ) if and and 1) ~+(KI,,,,)IJop 5 only if Z I , ~ ( K E~ Lz(r(Ep),I'(Eq)), ,~) in that case 11

h,T71(K1,WlIIHS )

=

(1 Kl,m

(IHS,

where &(X, 9)is the space of Hilbert-Schmidt operators from a Hilbert space X into another Hilbert space 9 of which the norm is denoted by (1 [ I HS.

38

Theorem 3.4 [14] Let p,q E R. For any Ei E L ( r ( E p )r,( E q ) )there exists a unique family of operators Li,,,, E L ( E f m ,E P ) , l , m 2 0 , such that m

which is called the chaotic expansion of 2, where the series converges weakly in the sense that 05

((54,+)) =

C ((IL,m(Ll,m)4>+)) ,

4 E r(Ep), 4 E r ( w .

l,m=O

Theorem 3.5 [I41 Let p , q E R. Given E E Lz(l'(Ep),r(Eq))let m

be the chaotic expansion. Then K L , E~ L 2 ( E f m , E P ) and the right hand side of (3.5) converges in L2(r(Ep), l?(Eq)).Moreover, we have m

m

4 Integral Representation of Hilbert-Schmidt Operators 4.1

Quantum Stochastic Integrals

A family { = : t } t & C L ( r ( E ) r, ( E ) * )of white noise operators is called a quantum stochastic process, where T C R+ is a (finite or infinite) interval. For a quantum stochastic process { E t } t G T , if t H ( (E& +)) is integrable on T for any d , + E r ( E ) and if there exists ZT E L(l?(E),r(E)*) such that ((ET4r $)) =

/

((st47

$)) dt,

4, '$ E r(E),

then the process { 5 t } t E T is said to be integrable on T . In that case, we write

and call it the white noise integral of

{Et}tGT

on T .

Lemma 4.1 Let {Et}tE~be a quantum stochastic process, where T C R is an interval. Assume (i) for any pair <, 17 E E , the map t

H

Et(<, 7) is measurable on T ;

(ii) there exist constants a 2 0, p 2 0, a locally integrable nonnegative function K E L;,,(T)

and a null set N

c T such that

l ~ t ~ < , ~ ~ l ~ ~ ~ ~ ~ ~E ,~s €~ E {, ~t E~T \l N< l

39

Then for a, t

E T,

the white noise integrals are defined:

I'

l-

a:Z,ds,

=,a,ds.

(4.1)

PROOF.It is straightforward by applying Theorem 3.1, see also [23].

I

As a simple application of Lemma 4.1, the white noise integrals: At =

a,ds,

A; = l a : d s

are well-defined and called the annihilation process and the creation process, respectively. Thus the white noise integrals in (4.1) are called the quantum stochastic integrals against the creation process and the annihilation process, respectively, and we write

If {Es}is an adapted process (see [6, 23]), we write

lI' dA:Z, =

Z,dA:.

For more study of white noise approach to quantum stochastic integrals, we refer to [23]. 4.2

Integral Representation

Let

E E L 2 ( r ( H )r,( H ) ) with the chaotic expansion W

z=

C

~

zl,m(Kl,m),

L~(H",H'"). ~

1E ,

I,m=O

Then for each 1, m

2 0 by the kernel theorem there exists ~

1 E H8' , ~ @I

Horn such that

Therefore, we write l,m=O

and consider

m

Mt

= EtSEt =

C

I~,m(lE~$mK~,m),

t 2 0,

l,m=O

where E t , t

2 0 , is the conditional expectation (see also [24], [25]) defined by @ = (F,) E B'

Et@ = (1;31C),

40

(4.2)

and then Et E C(Gp,Gp)for any t 2 0 and p E R. On the other hand, for each l,m 2 1 we have

Hence there exists a null set N

c R+ such that for all t E R+ \ N

Therefore,

Since for s E R+ \ N

we put

and E1,,-1(S)

= 0,

= &I,,(s)

sEN.

..

Then E J - ~ , ~and ( S )fi,m-l(s) satisfy the conditions (i) and (ii) in Lemma 4.1 and so the integrals

are well-defined, and EI,,,-l(s) and FI-~,,(S) are adapted. Hence by (4.3)we have

41

Let (4,) be a complete orthonormal basis for l?(H). Then for any p E R, {l?(e-PI)$,} is a complete orthonormal basis for Gp, where r(e-PI) is the second quantization of e-PI. Therefore, for each 2 E L 2 ( r ( H )T, ( H ) )and p , q 2 0 we have IlZl/HS;p,-q

= ll~(e-q~)2r(e-p~)llH~,

where I I ~ I I H S ; ~ , - ~ is the Hilbert-Schmidt norm of Z on L2(Gp,G-J and In particular, for any E BH G ~

IIl',m(Xl,m)II~s;p,-g

-

11 . 1 1 =~ 11 .~ [IHS;O,O.

cZpm -2q1 e 114,rn(X~,rn)IlkS

where s = ( ~ 1 , .. . , s l ) and t = ( t l , .. . , tm).Therefore, we have

Hence the maps

R + 3 s ++ E ( s ) =

CG,m-~(s),

R+ 3 s H F ( s ) =

C

&-I,~(S)

(4.6)

l,m

l,m

belong to L2(R+,L2(Gp,Lq) for any p , p > 0 and then from (4.2) and (4.5) we have

More generally, we have the following theorem

Theorem 4.2 Let p , q E R and let 2 E C2(Gp,Gq). T h e n there exist adapted processes {E(s)},?o and { F ( S ) } ~contained ~O in L2(R+,L2(Gp+,,Gq-s) f o r any r , s > 0 such that Mt = EtEEt admits the integral representation (4.7). Theorem 4.3 [4,261 Let {Z,}r=.=,and 2 be in L ( r ( E ) , r ( E ) * ) Then . En converges t o S in L(l?(E),r ( E ) * )i f and only zf the following conditions hold: (i)

z,(C,7) converges to z(c,7) f o r each t,7 E E .

(ii) There exist p

2 0, a > 0 and K > 0 such that lzn(t,q)l5 Kea(lElg+lol~), ( , 7 E E ,

n E N.

By applying Theorems 4.2 and 4.3 we have the following

Theorem 4.4 Let p , q E R and let 2 E L2(GP,Gq). T h e n there exist adapted processes (E(s)},>o and {F(s)},>o contained in L2(R+,L2(Gp+,,Gq-s) f o r any r, s > 0 such that E= (

( ~ 440)) ~ ,r +

J

m

E(s)dA, +

42

1

m

F(s)dA:.

(4.8)

Integrands in Integral Representation

4.3

The number operator N is uniquely specified by the symbol

@((, 7 ) = (E, 11) e(c,v),

t ,7 E E

and its Wick exponential wexp(-N) is defined by

In general, for two integral kernel operators Zl,,(K) and Zlt,m~(K') their Wick product or normal-ordered product is defined by

ZI,,(K) 0 ZV,d(K')= Zl+l',m+m'(Kk3 K'), see [5]. Let p , q E R and Kl,,,, E L ( E f m ,E?). Then from (3.2), (3.3) and (4.9) we have

where the series converges in L(r(E(pvq)+r), I?(&)) for any r > 0 with (2p'/2))/(-r10gp) For the proof, we refer t o [14]. Let E E. Then by Proposition 4.1 in [5] and (4.10) we have

< 1.

c

a where

*K

Il,m(Kl,m)u: = mI~,m-l(K~,m *

0,

I ,E~L(Efm,Ef('-') ) and Kl,m * c E L(Ef("-'),E Y ) satisfying

(c *

Kl,naE@m, P(1-1))

((K1,m *

oE@'"-", P )

63 c ) ,

=

(Kl,mpm,

=

(Kl,m(€@(m-i) 63 0, P ),

€3

17 E E .

On the other hand, if q m E Hol k3 Horn corresponds to K I , ~then , for almost all t E R+ 6t*Kl,m and Kl,,,,*bt are well-defined as operators corresponding t o K l , m ( t , .; .) and ~ 1 , ~ ( t., .), ; respectively. In that case, we write

DtI~,m(%n) = &1,m(61,m(t, .; .)). Then we have ( o t h , m ( Q , m ) * ) *= m I l , m - l ( K l , m ( . ;

t , .)).

Hence from (4.4)we have G,m-i(S) =

(DsEsI~,m(~l,m)*Es). , ~-I

s , ~ ( = S )DsEsIl,m(Kl,m)Esr

E R+ \N.

Therefore, for each 5 E LZ(I?(H),r(H)),the maps E ( s ) and F ( s ) in (4.6) are given by

E ( s )= (D,E,Z*E,)' ,

F ( s ) = D,E,ZE,.

(4.11)

Finally, by (4.11) and Theorem 4.4 we have the following theorem

Theorem 4.5 Let p , q E R and let 5 E L2(Gp,Gq).T h e n E admits the following integral representation: 3 = ((940, do)) I

+

1

W

(DsE3E*E8)*

43

References [l] K. Aase, B. 0ksenda1, N. Privault and J. Ubere: White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance, Finance Stochast. 4 (2000), 465496.

[2] S. Attal: A n algebra of non-commutative bounded semimartingales: square and angle quantum brackets, J. Funct. Anal., 124 (1994), 292-332. [3] S. Attal: Non-commutative chaotic expansion of Halbert-Schmidt operators on Fock space, Commun. Math. Phys. 175 (1996), 43-62.

[4]D. M. Chung, T. S. Chung and U. C. Ji: A simple proof of analytic characterization theorem for operator symbols, Bull. Korean Math. SOC.34 (1997), 421436. [5] D. M. Chung, U. C. Ji and N. Obata: Higher powers of quantum white noises in terms of integral kernel operators, Infinite Dimen. Anal. Quantum Probab. Rel. Top. 1 (1998), 533-559. [6] D. M. Chung, U. C. Ji and N. Obata: Quantum stochastic analysis via white noise operators in weighted Fock space, Rev. Math. Phys. 14 (2002), 241-272. [7] T. Hida: “Analysis of Brownian Functionals,” Carleton Math. Lect. Notes, no. 13, Carleton University, Ottawa, 1975.

[8] T. Hida: “Brownian Motion,” Springer-Verlag, 1980. [9] R. L. Hudson and K. R. Parthasarathy: Quantum Ito’s formula and stochastic evolutions, Commun. Math. Phys. 93 (1984), 301-323.

[lo] R. L. Hudson and J. M. Lindsay: A non-commutative martingale representation theorem for non-Fock quantum Brownian motion, J. Funct. Anal. 61 (1985), 202-221. [ll]R. L. Hudson, J. M. Lindsay and K.R. Parthasarathy: Stochastic integral representation of some quantum martingales in Fock space, in “From Local Times to Global Geometry, Control and Physics,” Proc. Warwick Symposium 1984/1985, pp. 121-131, Pitman RNM, 1986.

[12] U. C. Ji: Stochastic integral representation theorem for quantum semimartingales, J. Func. Anal. 201 (2003), 1-29. [13] U. C. Ji and N. Obata: Quantum white noise calculus, in “Non-Commutativity, InfiniteDimensionality and Probability at the Crossroads (N. Obata, T. Matsui and A. Hora, Eds.),” pp. 143-191, World Scientific, 2002. [14] U. C. Ji and N. Obata: A role of Bargmann-Segal spaces in characterization and expansion of operators on Fock space, J. Math. SOC.Japan. 56 (2004), 311-338. [15] U. C. Ji and N. Obata: Admissible white noise operators and their quantum white noise derivatives, to appear in “Infinite Dimensional Harmonic Analysis (H. Heyer, T. Kawazoe and K. SaitB, Eds.),” World Scientific. [16] U. C. Ji and N. Obata: Annihilation-derivative, creation-derivative and representation of quantum martingales, preprint, 2003. [17] I. Kubo and S. Takenaka: Calculus on Gaussian white noise I, Proc. Japan Acad. 56A (1980), 376-380.

44

[18] H.-H. Kuo: “White Noise Distribution Theory,” CRC Press, 1996. 1191 J. M. Lindsay: F e m i o n martingales, Probab. Theory Related Fields 71 (1986), 307320. [20] P.-A. Meyer: “Quantum Probability for Probabilists,” Lect. Notes in Math. Vol. 1538, Springer-Verlag, 1993. [21] N. Obata: An analytic characterization of symbols of operators on white noise functionals, J. Math. SOC.Japan 45 (1993), 421-445. [22] N. Obata: “White Noise Calculus and Fock Space,” Lect. Notes in Math. Vol. 1577, Springer-Verlag, 1994. [23] N. Obata: Generalized quantum stochastic processes on Fock space, Publ. RIMS, Kyoto Univ., 31 (1995), 667-702. [24] N. Obata: Conditional expectation in classical and quantum white noise calculi, RIMS Kokyuroku 923 (1995), 154-190. 1251 N. Obata: White noise appmach to quantum martingales, in “Probability Theory and Mathematical Statistics (S. Watanabe et al. Eds.),” pp. 379-386, World Scientific, 1996. [26] N. Obata: Wick product of white noise operators and quantum stochastic differential equations, J. Math. SOC.Japan, 51 (1999), 613-641. [27] K. R. Parthasarathy: ”An Introduction to Quantum Stochastic Calculus,” Birkhauser, 1992. [28] K. R. Parthasarathy and K. B. Sinha: Stochastic integral representation of bounded quantum martingales in Fock space, J. Funct. Anal., 67 (1986), 126-151.

45

THE DAWN OF WHITE NOISE ANALYSIS IZUMI KUBO Department of Environmental Design, Faculty of Environmental Studies, Hiroshima Institute of Technology 2-1-1 Miyake, Saeki-ku, Hiroshima 731-5193 Japan [email protected] Abstract

The author gives a historical overview of Brownian motion, Srstly. Then he is going to see how white noise analysis was established in time of its dawn. Further, some comments, which are related with primitive questions, will be stated.

$1. Short history of Brownian motion Titus Lucretius Carus (BC50 [l])is the first scholar who mentioned the importance of random movement of particles. He watched irregular movement of particles in sunbeam in dark house. By the observation, he could conclude the existence of atoms, which are invisible and move very rapidly and randomly’. In 1827, Robert Brown observed vital molecules contained in pollen which move actively in water under a microscope. He treated pollen grains of many kinds of flowers; Clarckia pulchella, (Enothera, Gramineae, Onagrariae, Asclepiadeae, Periploceae, Apocineae, Orchideae, Cryptogamous plants, Mosses, Equisetum, Equisetd and Phaenogamous plants etc. After him, we call the movement Brownian motion2. Such moving organic particles had been found already by Buffon, Needham and Spallanzani in 18th century as mentioned by himself and by Jean Perrin [8]. They thought that the vitality of particles causes the movement. However, Brown (1828 [2]) went on further. He examined various products of organic and inorganic bodies; for examples, the gum-resins, pit-coal, dust, soot, a fragment of window-glass, rocks of all ages, a fragment of Sphinx, aqueous rock, igneous rock, lava, travertine, stalactites, obsidian, pumice, manganese, nickel, plumbago, bismuth, antimony, arsenic and so on. Small particles, in which organic remains had never been found, moved in water randomly! Albert Einstein (1905 [4]), gave a kinetic theory of Brownian motion, the motion of small particles in liquid, firstly. Einstein’s model, which is so called Brwonian motion today, had been already studied by Louis Bachelier in the theory of speculation. His fluctuation-dissipation theorem proposed a method to determine Avogadro number. Marian Smoluchowski (1906 [5]) also investigated a similar formula.

lLouis George Gouy pointed out the worth of Lucretius’ atoms as referred in [8] ‘In this section, the author denotes the motion of Brown’s particle in liquid by bold style Brownian motion. Italic style Brownian motion means the stochastic process as its mathematical model established by Wiener and L6vy. The priority of research of the process belongs to L. Bachlier.

46

Perrin (1909 [7]) made experiments of various particles in various liquid, and got Avogadro number by Einstein’s formula. One of the best results was 6.88 x (the This is a fruit of statistical physics developed by Maxwel, newest one is 6.0221367x Boltzmann and others, and the victory of Atomistik against Energitik. Brownian motion as a stochastic process was firstly investigated by Louis Bachelier (1900 [3]) as thesis of financial theory for the doctorate in mathematical sciences. His speculation model assumes that logarithmic stock prices are exchanged by random walks. He studied also Brownian motion as their limit process. He discussed the non-differentiability of its paths and other properties without modern probability theory, which was established later. Norbert Wiener (1923 [9]) and Paul L6vy (1940 [la]) constructed Brownian motion and researched in the framework of modern probability theory. In particular, each of them constructed Brownian motion and established analysis of functions of Brownian motion by his own way. Wiener gave a method of analysis of non-linear functions of Brownian motion, which is so called Wiener chaos (1938 [ l l ] ) . Kiyosi It6 (1944 [l3]) wrote the his most famous article on stochastic integral, which serves very important basic theorems in many fields including mathematical finance theory started by Bachelier. He gave also a very beautiful and useful theory of Wiener-It6 expansion (1951 [14]). For a physical model of motion of a particle in liquid, Bachelier-Einstein’s Brownian motion does not take resistance force of liquid into the equation. Considering Stoke’s force, Paul Langevin (1909 [6]) gave an equation of the velocity of Brownian motion and Gorge Eugene Uhlenbeck and Leonard Salomon Ornstein (1930 [lo]) discussed the corresponding process. They are so called Langevin equation and Ornstein-Uhlenbeck’s Brownian motion, respectively. Since Stoke’s force can be applicable only to the case that a particle moves with constant velocity, Ryogo Kubo (1966 [Zl]) used Boussinesq’s force with memory and his linear response theorem. His formula fits to the experiment of Brownian motion by Kohji Obayashi, Tomoharu Kohono and Hiroyasu Uchiyama (1983 [29]). In the following section, the author will see the root to white noise analysis with over view of several key theories. $2. Hida’s research before 1975

In 1975, the new approach of white noise analysis was proposed by Takeyuki Hida. How did he get its idea? Here some of his results, which consist the basis of the theory3, will be picked up. White noise is originally understood as the derivative of Brownian motion. Bachelier asserted and LBvy discussed precisely, non-differentiability of Brownian motion. It is interesting that Hida also showed a precise continuity of Brownian motion with multi-parameter in his early paper (1958 [15]). By the reason of non-differentiability, a rigorous treatment of white noise by Izrail’ Moiseevich Gel’fand’s generalized process [17] is necessary. The use of reproducing kernel space is one of his important idea. He mentioned it already in 1960 for the study of canonical representation problems. In 1967, he and Ikeda 3Many papers referred here are selected in [30].

47

applied it to study generalized process. It allowed a free calculus on non-linear functionals of white noise. The study of infinite rotation group was begun in 1964 and was discussed in terms of white noise analysis later. The following notations will be used often later. Let Eo be a Hilbert space with norm 11 . 110 and E C Eo c E‘ be a nuclear triplet. Then Bochner-Minlos’ theorem (see [17]) guarantees the existence of Gaussian measure p on E’ with characteristic functional 1 C ( ( ) = e~p[--JI(l(~],( E E . The measure p is called the standard Gaussian measure. 2 The infinite dimensional rotation group O(E’) is defined by

O(&’) = {h’ : h is a homeomorphism of &, llhfllo = llfllo for f E E } , where h* is the dual homeomorphism of h defined by (h“z,.$) = (z,he) for ( E E . Then h* E O(&’) preserves the measure p and gives a unitary operator uh on (L2)= LZ(&’,p ) by (Uhf)(Z) = f ( h * z )for z E E‘. For simplicity we note O(w) = O(&’) and call it m-rotation group.

1. Properties of Paths Hida proved the upper and lower estimates for uniform continuity of N dimensional LBvy’s Brownian motion (1958 [IS]).

Theorem A. Let { B ( x ) ,x E RN} be Liwy’s Brownian motion and put cpc(r) = { r ( 2 N (log r ( + clog I log rl}1/2. Then (2) If c > 8N

+ 1, for almost every w,there exits a positive number p = p(w) such that r = )x- y1

< p implies JB(x)- B(y)l _< cpc(r)

(ai) If c < 1, for almost every w , there exits a pair (z, y) for any p > 0 such that lB(x) - W Y ) I

> cpdr)

(0 < r = Ix - yI

< p).

2. Canonical Representation A stationary Gaussian process X ( t ) can be represented in the form

X ( t )=

J”

-m

k(t - U ) d B ( U )

by stochastic integral of Brownian motion { B ( t ) : t E R} under a suitable condition. Hida (1960 [IS]) showed that reproducing kernel Hilbert spaces are useful for studying representation and causal analysis of Gaussian processes, which are linear functionals of Brownian motion. Let { X ( t ) }be a continuous Gaussian process of mean 0 with the covariance function r(t,s) and M t ( X ) = linear span of { X ( s ): s 5 t } . We see his results for the case that M t + h ( X )= M , ( X ) and that n M , ( X ) = ( 0 ) .

n

h>O

t

48

Theorem B. There exist Gaussian random measures {B(’)(t)}independent of each other and kernels {F,(t,u)}such that

For the analysis, he used the reproducing kernel Hilbert space 31 = 31(r)with the reproducing kernel r ( t , s ) .Put at = N t ( r )= linear span of {r(s,.) : s 5 t } and let E ( t ) be the projection form 31 onto 3tt. Applying Hellinger-Hahn theorem to the resolution of unit { E ( t ) } ,he obtained the theorem. Prediction theory is its application. 3. Projective limit of spheres Hida and Hisao Nomoto (1964 [18])discussed the measure space given by the projective limit of spheres with uniform measures. Let S, be the n-dimensional sphere given by x: +xi +. . . + = n + 1 and p,, be the uniform probability measure of S,. For m > n, define projection rn,, by

..

rn,rn(Zl>~ 2 , .

zrn+~) =

n f l

& + Z; + . . . +

( X I , 22,..

., xn+l)

x:+1

Then we have a projective system {(Sn,p,,,~n,rn)}. Let us denote by (Sm,p,) its projective limit space and let r,,be the projection from S, onto S,. They showed

Theorem C. (Sm,p,)

is a standard Gaussian measure space; that is, {&(x) = x,} is a independent Gaussian sequence subject to N ( 0 , l ) f o r x = ( X I , 22,.. . ,xn+l,. . .) E S, with respect t o p,.

Rotational flows on S, approximate flows on S,, which are called finite dimensional flows and are members of m-rotation group (Nomoto 1966, 1967 [19]). Further, HidaNomoto (1967 [20]) constructed band limit Gaussian processes by the approximation. 4. Non-linear Analysis For non-linear analysis of Gaussian and Poisson processes, Hida and Nobuyuki Ikeda (1967 [22]) presented a method by means of reproducing kernel Hilbert spaces in Berkley Symposium. They showed that the Wiener-It6 decompositions are given very naturally. In the paper, they used theory of Laurant Schwartz’ distributions or generalize functions of Gel’fand [17] successfully. Since this fact is directly related to white noise analysis, here its brief sketch is given. We observe Schwartz’ S, S‘ and the standard Gaussian measure p. Then, in the sense of Gel’fand’s generalized process, Brownian motion is identified with B(7) = /q(u)dB(u) = ( G V ) , 11 E S. Their first point is that C(c - 7) is positive definite and can be a reproducing kernel of a Hilbert space 3. The second is that functional derivatives operate on the Hilbert space and that Taylor expansion corresponds to Wiener-It6 decomposition. The 7-transform4 40riginally, 7 was denoted by

T.

49

defined by

is an isomorphism from (L2)= L2(E',p ) onto F.The Taylor expansion C(E - 7))

[

= exp -

;

KJE, 7 ) =

m

$ - 111121 = p=o cC(E)KP(OC(17)7 1

(/E(t)S(t)q

corresponds to Wiener-It6 decomposition. Let Fp be the Hilbert space with reproducing kernel K,(J,v). For f in a symmetric L2-space E2(Rn),I p ( f ) is the multiple Wiener integral o f f :

Ip(f)= / . . . / f ( ~ , u,.z. . , . " ) d B ( ~ , ) d B ( ~ z ) d B ( u n ) . with

Put Jp(f; E ) = iPC([)I;(f : E )

I,*(f;E ) =

1.. .

f(h ,t z , . . . ,t p ) E ( t i ) E ( t z ) . . . E(tp)dtidtz . . . d t p E

Set 31, = Ip(EZ(Rp)) c 31 and Fp = Jp(E2(Rp))c F.

Theorem D. Ez(Rp) is isomorphic to 31, and F b the isomorphisms Ip(.)and Jp(.),

p,Y

respectively. T h e orthogonal decompositions ( L z )=

c@

c @ Fp m

31, and F =

p=o

hold.

p=a

It is important that both direct sums are realized by sums of functions: m

9 b )=

c

PP(4

c m

and

( 7 9 ) ( 0=

JP(fP)(E)?

p=o

p=o

and the Taylor expansion of ('Ty)(E)exp[fllElli] corresponds to its orthogonal decomposition. Why we need S (S') and how differentiability of basic functions in S can be use? Is the necessary property only the nuclearity of linear spaces? Answer was given his new approach.

5. Infinite Dimensional Rotation Group Hida-Kubo-Nomoto-Yosizawa (1969 [23]) wished to realize LBvy's projective invariance of Brownian motion as point wise transformations on a basic measure space. For the realization, they introduced a testing function space Do of special decay order at 03, which is nuclear with a-Hilbertian norms :

'The author introduces J , ( f ; E ) , which was not used in Hida's paper, only for convenience of explanation.

50

For E = DO, we have the co-rotation group O(D6). Each g =

(:

i)

E SL(2, R) acts as a linear homeomorphism gcp defined by

which preserves Lz-norm [I .110. Therefore, its dual homeomorphism g* belongs to O(Db). A natural Brownian motion defined on the probability space ( D b , p ) satisfies point wise projective invariance. Further, each one-parameter subgroup of SL(2, R) gives a whisker in co-rotation group O(Db). Flows of Brownian notion and Ornstein-Uhlenbeck process are such examples. Hida (1970 [24], 1972 [25], 1973 [26]) investigated infinite dimensional Laplacian and harmonic analysis relating to infinite dimensional unitary operators arising from O(E').

$3. White Noise Analysis Hida (1975 [27]) gave a series of lectures in Carleton University and published a lecture note which is the first stage of white noise analysis. The author was informed the theory by Hida himself in Erlangen (1976) and understood its importance immediately. The new idea is that a generalized process X ( t ,w ) is considered as a generalized functional in w for fixed t , not a distribution in t for fixed w . As well known, the idea gives us much freedom to analyze Brownian functionals and can be applied to many fields in stochastic analysis. Basic concepts are as follows. Let p be the standard Gaussian measure on S'. Let 7 and Jp be the transformations stated in subsection 4 of 52. Observe the inclusions G(n+1)/2(Rn) C Z2(R'') C g-("+')l2(Rn)of symmetric Sobolev spaces. The inclusions can be transformed6 to 3p) c Fn C FA-") by J,. Put 'lip) = 7-'F?), 31, = T-'Fn and Xi-") = 'T-'Fi-n), then we have inclusions: m

CQ

m

(Lz)+=c$3-1~)c(Lz)=c$31nc(Lz)-=c$31~-"). n=O

Hida derivative

aB(t)

n=O

n=O

is defined7 by

Brownian motion B ( t ) is considered as B ( t ) = Z,(X[O,~~)E (L') for t 2 0. The derivative 1 B(t)= h-tO lim -(B(t + h) - B ( t ) )= ZP(&) in (L2)h has meaning in the sense of generalized Brownian functional; that is, an element of ( L z ) - . This is an answer to the question asked for Hida-Ikeda's paper in subsection 4 of 52. The 6The transformations Jp and 7-' must be extended properly. 7The original definition of Hida derivative was given for each term of Wiener-It6 decomposition

51

differentiability of test functions guarantees the differentiability of Brownian motion in t for each fixed w . In the frame work, one can treat L6vy’s Laplacian, Laplace-Bertrami operator, Feynman path integral etc. well. Later, Kubo-Takenaka modified (1980 [28]) 7-transform to S-transform slightly:

where (E’) is defined below. By S-transform, Hida derivative is described as

The relation of two definition is clear by observing the analytic continuation from the imaginary axis to the real axis. They constructed functional spaces by using reproducing kernels fully just as Hida’s idea. Let KO be the reproducing kernel Hilbert space with kernel exp[((,v)]. Then A nuclear triplet (K)c KO c (K’)can be introduced by using the Hilbertian norms of the basic space E (see [28]). Since S ( L z ) = KO,one can define triplet ( E ) c ( L 2 )c (E’) as the inverse image of the triplet by S-transform. $4. Comments

The author had spent with Hida in Nagoya University from 1967 to 1983 and watched the creation of white noise analysis, which might be called Hida Calculus. The following comments are based on questions in his mind in those days. 1. O(c0) In Hida and Nomoto’s projective limit of spheres, SO(n) acts on S, naturally. We call Oj(S,) = U,SO(n) the finite dimensional rotation group of S,. The closure of Oj(c0) is not so clear. Of course, Oj(c0) is a small group of really finite rotation. However its closure under a suitable topology is big enough. For example, the shift transformation of iid Gaussian random variables is really infinite dimensional. It can be approximated by elements (g2,) of u,SO(n), where 92, rotates as 92, : (z1,z2,.. . ,~ 2 +~ ( Q ) ,z~ xs,x2, , 27,2 4 , . . . , X Z , , ~ ~ - 1 )This . idea may be extended to any g E O(c0). 2n + 2n - 2 -+ ... -+ 4 -+ 2 + 1 -+ 3 -+ 5 + ... -+ 2n - 1 4 (2n)

t

-1

2. Sobolev Norms Hida introduced Sobolev norms to control (Lz)+C ( L 2 )C ( L 2 ) - . The norm fn(u)on R” is defined by

52

llfnlln

for

where fn(v) is the Fourier transform of fn(u). By @, we denote the space of functions with finite norm. The function space is convenient to treat trace. Set

m

m

m

(L')+ =

C ex?), II'PII' = C c~IIJo~JIEfor n=O n=O

'P =

C pn?

n=O

' ~ E n

.tit'

by a suitable increasing sequence {c,,}. This norm adjusts well to the trace properties of kernel functions fn E En. But it has some difficulties for calculus. For example, it is hard to see that exponential function belongs to (L')+ or not and that for y, I,/J E (L2)+does yI,/Jbelong to (L')+ or not. Its modification works sufficiently. We need to research more Sobolev type norms.

3. Generalized Random Variables CP E (E') may be called a generalized random variable. If CP E (L'), then @ is a usual random variable. We say that CP E (E') is ordinal, if CP E (L') = L'(&', p). (i) Is there any useful criterion for that CP is ordinal? (ii) Suppose that {an}and @ are ordinal and and an converges to @, Is there any criterion for that CP, converges to @ in distribution sense? (iii) Suppose that {@,} and (qn}are ordinal, and that @, + @ and an+ Q as n Is there any criterion for that the product converges in (€').

+ 03.

(iv) Can we give any natural definition of the product @* using (iii) for suitable class of generalized random variables? 4. Renormalization

Hida introduced a scheme of renormalization as follows. Let us observe Brownian motion on the interval [0, TI. For the classical path y(t) = y(t,z, a ) from (0, T) to (T,a ) , define a perturbed path

by Brownian bridge. He expected that the propagator with potential V is given by

The renormalization process for : : is given by exP ! !!! &Yo Ak - 7~i V(Ya(tk--l))& 2fi tk
[

') (a,

IAlW

AkYa

53

for partition A : t o = 0 < tl < t2 < . . . < t, = T , ( A(= max{tk - tk-l : 1 5 k 5 n}. The author prefers this process and hopes for some one to research in this direction. One of the purpose of Kubo-Takenaka ([as]) was to treat it in their framework. This approach is deeply related to the problem in previous subsection. 5. Path Hida and Si Si are studying variational calculus of random fields. Its origin is found in investigation of LBvy’s Brownian motion B(x) with multidimensional parameter. Let us suppose that the parameter space is R3. Let C ( t )be a sphere with radius t and center xg. Further the origin 0 is outside of C(t). A method of interpolation B(x) of B(x) by the information of B(.) in germ of C ( t ) .We can give 6(x) by

d with smooth kernels KO, K1 and the normal derivative symbolically. But B ( z ) is not an’ differentiable in path level as seen in Theorem A. However the following expression holds rigorously in the usual probability theory:

-

2

B(x) = -t Integral of B ( y ) on the sphere C ( t ) with smooth kernel becomes smooth in t. The differentiability of such kind was observed in LCvy’s M(t)-processes as discussed also in [16l. There are several questions: (i) Do paths of the Brownian motion have such properties generally? Is it possible to show that by some precise discussion on path similar to Hida [15] ? (ii) What is the characterization of a function space of such functions? (iii) In the framework of white noise analysis, can we show that integrals of a B ( y ) (in Hida’s sense) on smooth surfaces are ordinal random variable? (iv) More generally, do we have any good criterion for that an element ordinal random variable?

E (E’) is an

Acknowledgement The author thanks to the organizer of the international conference for giving him the chance of this article, particularly to Professor Hida, who had guided him to this interesting field in the dawn of white noise analysis. The author also thanks to Masuyuki Hitsuda, who taught him an essay of Torahiko Terada in which Lucretius’ molecule is described.

54

References [l]Titus Lucretius Carus; De Rerum Natura, (BC 50). (Nature of things, translated by William Ellery Leonard', Dover Publications, Incorporated (2004))

[2] Robert Brown; A brief account of microscopical observations made in the months of June, July, and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and Inorganic bodies, The Philosophical Magazine and Annals of Philosophy, Series 2, 4. No.21 (1828), 161-173. [3] Louis Bachelier; Thhorie de la spkulation. Annales Scientifiques de 1'Ecole Normale SupBrieure, 111-17 (1900), 21-86. Thesis for the Doctorate [4] Albert Einstein;Uber die von der molekularkinetischen Theorie der Warme gefordete Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen. Annalen der Physik, 17 (1905), 549-560.

[5] Marian Smoluchowski; Bulleitn de L'Acad. des Sc. de Cracovie. Juillet, Libraire FBlix Alcan (1906) [6] Paul Langevin; Sur la theorie du mouvement brownien, C. R. Acad. Sci. (Paris) 146 (1908), 530-533

[7] Jean Perrin; Movement Brownien et Rkartik Molkculaire, Annales de Chimie et Pyisique, 8"" sBries, Septembre (1909)

[8]Jean Perrin; Les Atomes. Felix Alcan, Paris, Nouvelle collection scientifique, directeur Emile Bore1 (1912) [9] Norbert Wiener; Differential-Space, J. Math. and Phys., 2 (1923), 131-174 [lo] Gorge Eugene Uhlenbeck and Leonard Salomon Ornstein; On the theory of Brownian motion, Physical Review 36 (1930), 823-841. 1111 Norbert Wiener; The homogeneous chaos, Amer. Math. J. , 60 (1938), 897-936

[12] Paul LBvy; Le mouvement brownien plan, Amer. J. Math., 62 (1940), 487-550. [13] Kiyosi It6; Stochastic Integral, Proc. Imp. Acad. Tokyo 20 (1944), 519-524. [14] Kiyosi 1 6 ; Multiple Wiener Interal, Journ. Math. SOC.Japan 3 (1951), 1-51.

[15] Takeyuki Hida ; On the uniform continuity of Wiener process with amultidimensional parameter, Nagoya Math. J. 13 (1958) 53-61. [16] Takeyuki Hida; Canonical representations of Gaussian processes and their applications, Memoirs Coll. Sci., Univ. Kyoto A33 (1960) 109-155. 'This translation is available from the web site - htpp://classics.mit.edu/Carus/nature-things.htm/

55

[17] Izrail’ Moiseevich Gel’fand and Naum Jakovlevich Vilenkin ; Generalized function 4, Some applications of harmonic analysis. Equipped Hilbert spaces, (1961) [18] Takeyuki Hida and Hisao Nomoto; Gaussian measures on the projective limit space of spheres, Proc. Japan Acad. 40 (1964) 301-304. [19] Hisao Nomoto ; Finite dimensional approximations to some flows on the projective limit space of spheres. I, II., Nagoya Math. J. 28 (1966),167-177. ibid. 29 (1967), 127-135. [20] Takeyuki Hida and Hisao Nomoto ; Finite dimensional approximation to band limited white noise, Nagoya Math. J. 29 (1967), 211-216. [21] Ryogo Kubo; The fluctuation-dissipation theorem, Rep. Prog. Physi. 29 (1966), 235 [22] Takeyuki Hida and Nobuyuki Ikeda; Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral, Proc. 5th Berkeley Symp. on Math. Stat. Probab. 2 (1967), 117-143. [23] Takeyuki Hida, Izumi Kubo, Hisao Nomoto and Hisaaki Yosizawa; On projective invariance of Brownian motion, Publ. RIMS Kyoto Univ. A4 (1969), 595-609. [24] Takeyuki Hida; Note on the infinite dimensional Laplacian operator, Nagoya Math. J. 38 (1970), 13-19. [25] Takeyuki Hida; A probabilistic approach to infinite dimensional unitary group, Proc. Japan-USSR Probab. Symp. (1972) 66-77. [26] Takeyuki Hida; A role of Fourier transform in the theory of infinite dimensional unitary group, J. Math. Kyoto Univ. 13 (1973) 203-212. [27] Takeyuki Hida; Analysis of Brownian Functionals, Carleton University, Carleton Math. Lecture Notes 13 (1975). [28] Izumi Kubo and Shigeo Takenaka; Calculus on Gaussian white noise. I, 11, I11 & IV, Proc. Japan Acad., 56A (1980), 376-380, ibid. 56A (1980), 411-416, ibid. 57A (1981), 433-437, ibid. 58A (1982), 186-189. [29] Kohji Obayashi, Tomoharu Kohono and Hiroyasu Uchiyama; Photon correlation spectroscopy of the non-Markovian Brownian motion of spherical particles, Physical Review A, 27, No.5 (1983), 2632-2641. [30] Takeyuki Hida; Selected papers of Takeyuki Hida, Eds. L. Accardy, H.-H. Kuo, N. Obata, K. Saitb. Si Si and L. Streit (2001).

56

WHITE NOISE STOCHASTIC INTEGRATION

HUI-HSIUNG KUO Department of Mathematics Louisiana State University Baton Rouge, LA 70803, USA E-mail: kuoOmath. lsu. edu White noise is often regarded as the informal nonexistent derivative B ( t ) of a Brownian motion B ( t ) . Before K. It6 introduced the stochastic integral in 1944, white noise had been used as a random noise which is independent at different times and has large fluctuation. It was an innovative idea of It6 t o consider the product of white noise B ( t ) and the time differential d t as a Brownian motion differential dB(t), a quantity to serve as an integrator in the It6 theory. In 1975 T. Hida introduced white noise theory which provides a rigorous mathematical definition of B(t) as a generalized function defined on the space of tempered distributions on the real line. The white noise B ( t ) can further be regarded as a multiplication operator and B(t) = 6’t + 6’; with 6’t being the white noise differentiation and 6’; its adjoint. In this paper we will give a brief survey of Hida’s theory of white noise and its applications to stochastic integration. We will use the operator 6’; t o study stochastic integrals with anticipative integrands and stochastic differential equations with anticipative initial conditions and integrands. We will also point out and describe several perspectives for further applications of white noise theory to stochastic integration.

1. Brownian motion and white noise

Let B(t) be a Brownian motion. Its informal derivative B ( t ) , which does not exist in the ordinary sense, is called a white noise. We give three simple examples of differential equations to motivate the need to establish the theory of white noise.

Example 1.1. Consider the differential equation with white noise B(t)

dXt - - a x , +B(t),

xo = zo, dt where a E R. By regarding this equation as an “ordinary” differential equation, we can derive the informal solution X t = zOeat +

t

e a ( t - s ) ) B ( s )ds.

57

But what is the integral s,' e"(t-s))B(s) ds? Observe that by the mean value theorem n

n

i=l

i=l

which converges in L2(S2)to a Gaussian random variable with mean 0 and variance

so t

Thus we can interpret the informal integral e"(t-s)B(s)ds as a Wiener t integral e"(t-s) d B ( s ) . Then the solution X , is given by

so

X t = zOeat

+

Jd

t

ea(t-s) d B ( s ) .

Example 1.2. Consider a differential equation with noise B ( t ) B ( t ) dXt = ax, + B(t)B(t), xo = 20. dt Again, by regarding this equation as an "ordinary" differential equation, we can derive the informal solution

Xt = zOeQt +

l

ea(t-S)B(s)B(s) ds.

(1)

But what is the integral s," e"(t-s)B(s)B(s)ds? There are two ways to interpret this integral. The first way is to combine B ( s ) and ds together and regard B ( s )ds = d B ( s ) as an integrator for It6 integral17. Then the solution is given by

X t = zOeat +

jDt

e"(t-S)B(s) dB(s).

The second way to interpret the integral s," e"(t-S)B(s)B(s) ds is to regard B ( s ) as an operator 8: in the Hida theory of white noise12y13>25132. Then the solution is given by

Xt = xgemt+

t

8; (e"(t-s)B(s))ds.

(2)

We will explain the operator 8; in Section 2 and define the white noise stochastic integral s,' 8 , " ( f ( s ) ds ) in Section 4. 58

Example 1.3. Consider a linear stochastic differential equation dXt = a ( t ) X t d t

+ P(t)Xt dB(t),

X O = ZO,

(3)

where a(t)and P ( t ) are deterministic functions. There are two methods to solve this equation.

Method 1 (It6 theory of stochastic integration). We need to guess that the integrating factor

Then use It6’s formula to derive the solution

Method 2 (Hida theory of white noise). Rewrite Equation (3) as dXt = a ( t ) X t d t

+ 8;(,B(t)Xt)d t ,

Xo = zo .

(5)

Then take the S-transform Ft(() = (SXt)(<) which , we will explain in Section 2, to get the following ordinary differential equation for each test function <:

d -Ft(E) = a(t)Ft(E) E(t)P(t)Ft(E), Fo(E)= Zo. dt The solution is given by (there is no need to guess as in Method l!)

+

Then use the following fact from white noise theory

to get the same solution of Equation (3) as given in Equation (4). In the above examples the equations and solutions have different degree of involvement with the white noise B(t). In Example 1.1the equation is solved as an “ordinary” differential equation and the solution requires the concept of Wiener integral. In Example 1.2 the equation is again solved as an “ordinary” differential equation, but the solution is interpreted as either an It6 integral or a white noise integral. In Example 1.3 the equation is interpreted as either a stochastic differential equation of It6 type or a 59

white noise stochastic differential equation, while the solution is derived by different methods. White noise had been used informally before K. It617 introduced the theory of stochastic integration in 1944, in which B(t)and dt are combined together to form dB(t) as in integrator. In 1975 T. Hida12 introduced the theory of white noise so that B(t)itself is a rigorous mathematical object. The progress can be illustrated by the following diagram:

- -

“B(t)”

K. It6

dB(t)

1944

T. Hida

1975

AMS

B(t) ------+ 60H40 2000

One of the many motivations for T. Hida to introduce the theory of white noise is to extend the It6 theory of stochastic integration. In this paper we will briefly review the development in this direction and point out several perspectives for further investigation. 2. White noise theory Let S(R) denote the Schwartz space of rapidly decreasing functions on R. The dual space of S(R) is denoted by S’(R).It is well-known that S(R) is a nuclear space and we have a Gei’fand triple S(R) c L2(R) C S’(R). Let p be the Gaussian measure on S’(R) with the characteristic function

where I . 10 is the L2(R)-norm. Then (S’(R), p ) is a probability space. The stochastic process B(t,z) = (z, l p t l ) , t 2 0, z E S’(R),is a Brownian motion. Then we have B(t) = z(t). Thus the elements of S’(R)can be regarded as the “sample functions” of white noise. For convenience, let ( L 2 )denote the complex Hilbert space L2(S’(R),p ) . By the Wiener-It6 decomposition theorem, each ‘p in (L2) can be uniquely decomposed into the sum

E 21(R”), n=O

where In is the multiple Wiener integral of order n with respect to the above Brownian motion B(t) and 2p(Wn) is the complex Hilbert space of symmetric L2-functions on R”. Moreover, the (L2)-norm of is given by 1 0 0

60

For each p 2 0 and cp given by Equation (6),define \

where A is the operator A = -d2/dx2 can introduce the following spaces

112

+ x 2 + 1. By using these norms we

(S,)= {‘p E ( L 2 ); IIcpllP < 0 0 ) ( S ) = projective limit space of

{ (S,) 1 p 2 O}

(the space of test functions) (S,)*= dual space of (S,)

(S)*= inductive limit space of { (S,)* 1 p 2 O} (the space of generalized functions)

It is a factz5 that (S) is a nuclear space and so we have a Gel’fand triple ( S ) C ( L 2 ) c ( S ) * . If E &(a), the complexification of S(W), then the

<

function

belongs to the space (S). The S-transform of a generalized function @ in (S)* is defined to be the function

(s@)(<) = ((@,:e(’zt):)),< E sC(a), where ((., -)) is the bilinear pairing of (S)*and (S). The next theorem characterizes generalized functions in the space (S)* in terms of their S-transforms.

Theorem 2.1. (P~tthoff-Streit~~ 1991) A complex-valued function F on of a generalized function in (S)* if and only i f it satisfies the following conditions:

Sc(W)is the S-transform

<,q z E c.

(1) For any of

E Sc(W),the function

F(z<+ q) is an entire function

( 2 ) There exist constants K , a , p 2 0 such that

I F ( J )I I Kealtl:,

V < E S~(IW>.

There are other spaces of generalized functions and the corresponding characterization theorems, see the references in the bookz5 and the survey paper~~~J~. 61

Let y E S'(R). For

'p E

(S) represented by Equation (3), we define 00

n=l

The operator Dy, differentiation in the direction y, is a continuous linear operator from (S)into itself. The adjoint Dj is a continuous linear operator from (S)* into itself. In particular, when y = &, we have the operators Da, and Di,, which are often denoted by at and a,*, respectively. The operator at is called the white noise differentiation operator. For y E S'(R), the multiplication operator =

Qy

(a,

Y)V,

'P E

(S).

+

is a continuous linear operator from (S) into (S)* and Qy= Dy Dj.In particular, for y = &, we have B(t) = at 8: as continuous operators from ( S ) into (S)*. Moreover, the following commutation relationships hold:

+

[as,at] = 0, [a,.,a;] = 0, [as,a;] = &(t)'. The S-transform of

DE@ is given by

S ( q y ) ( E )= ( Y , W @ ( O ,

E E Sc(R).

In particular, when y = &, we have

s(a;@)(E)= E(t)S@(E), E E SC(R), which is very useful for white noise stochastic integrals. On the other hand, the S-transform of at'p for 'p E (S) is given by

d

s(at'p)(E)= &WE + ZSt)l,,O'

E

E

Sc(R).

The S-transform is not so useful to handle the white noise differentiation at'p of a test function 'p in (S).

3. White noise stochastic integrals

A function @ : [a,b] (S)* is called Pettis integrable if for any 'p E (S), the function ((a(.),'p)) is measurable and belongs to L1[a,b]. In that case, there exists a unique Q E (S)* such that .--)

b

(P,cp)) =

J ((@(tL'p))dt,

'p

E (S).

The unique Q, denoted by J: @ ( tdt, ) is called the Pettis integral of @(t).

Theorem 3.1. Suppose @ : [a,b] -+ (S)* satisfies the conditions: 62

<

(a) S@(.)(<)is measurable for any E S,(R). (b) There exist constants K , X,p 2 0 such that

I”

Is@(t)(<)ldt5 KeAIcIi,

v< E S,(R).

T h e n @ ( tis ) Pettis integrable and

A function 4? : [a,b] 3 (S)*is said to be Bochner integrable if it satisfies the conditions: (a) For any cp E (S),the function ((@(.),cp)) is measurable. (b) Thereexistsp 2 0 such that a(.) E (Sp)*a.e. and l ( @ ( . ) ( I p

E L1[a,b].

In that case, there exists a unique Q E (S,)*such that

1

b

((Q, cp)) =

The unique Q, denoted by

((@(t), cp))

cp E ( S ) .

Jt @ ( tdt,) is called the Bochner integral of @(t).

Theorem 3.2. Suppose @ : [a,b] + ( S ) *satisfies the conditions:

<

(1) S+(.)(<) is measurable f o r any E Sc(R). ( 2 ) There exist constants A, p 2 0 and a nonnegative integrable function L o n [a,b] such that

IS@(.)(<>I 5 L(.)eXIElz, V <

E

S,(R).

T h e n @(t)is Bochner integrable and

for any q > p such that 2Xe2((A-(4-p)((&< 1.

If @ : [a,b] --+ ( S ) *is Pettis integrable, then the function t

++

a:@(t)is

also Pettis integrable and

Moreover, If @ : [a,b] 4 (S)* is Bochner integrable, then the function t H a:@(t)is also Bochner integrable and Equation (7) holds. 63

4. Hitsuda-Skorokhod integrals In Example 1.2 we interpreted the multiplication by B(t) in Equation (1) as the operator a,* in Equation (2). In Example 1.3 we interpreted the integrator d B ( t ) in Equation (6) as 8; dt in Equation (5). The reason for this interpretation is given by the following theorem.

Theorem 4.1. (Kubo-TakenakaZ31981) Let p(t) be nonanticipating and J” Il‘p(t)IIzdt < 00. Then the function t H d;‘p(t) is Pettis integrable and

6” (‘pW 1

b

a,*

dt =

d t )d B ( t ) ,

where the right-hand side is the A S integral of Observe that in the white noise integral

‘p(t).

J: 8; (cp(t))dt there is no need

to require that the stochastic process ‘p(t> to be nonanticipating. Therefore,

it provides an extension of the It6 integral. However, the extension would be more meaning if the integral Jf ( ‘ p ( t ) ) dt defines a random variable instead of just a generalized function in (S)*. Recall a fact from the book^'^^^' that W S ’ ( W , p ) c (S)*. Thus we make the next definition.

a,*

ul
Definition 4.1. By a Hitsuda-Skorokhod integral we mean a white noise integral a,*(‘p(t))dt when it defines a random variable in LP(S’(R),p ) for some 1 < p 5 00.

s,”

The integral J : 8; ( ‘ p ( t ) ) dt was introduced by Hitsuda14 in 1972 and by S k ~ r o k h o din~1975 ~ by different methods and notations.

Jt

Example 4.1. d,*B(l)dt = B ( l ) 2 - l . On the other hand, the extension by It618 gives a different value B(1)d B ( t ) = B(1)’.

Jt

-

Theorem 4.2. Suppose ‘p : [u,b] (S)*satisfies the condition that Jf II(N 1)1/2‘p(t)llidt< 00, ( N : number operator). Then S,ba,.io(t)dt is a Hitsuda-Skorokhod integral and

+

1l

8t*‘p(t> dt$ =

l

Il‘p(t>ll2,dt +

J” J” ((at(P(4,

a8‘p(t>>>, dsdt,

a

a

where ((., .))o is the inner product o n (L’). Moreover,

64

5. Extensions of It6's formula Let 8 be a C2-function. ItS's formula states that

+I t

8 ( B ( t ) )= O(B(a))

e'(B(s))d B ( s )

a

+

The white noise formulation of this equality is given by

+I t

q q t ) )= e ( B ( a ) )

a:el(B(s))ds

a

+l

l

e l p ( s ) ) ds.

In this formulation, It6's formula can be generalized to Hitsuda-Skorokhod integrals and generalized functions.

Theorem 5.1. ( H i t s ~ d a '1972) ~ ~ ~Suppose ~ 9(z,y ) is a C2-function and assume that the functions

all belong to L 2 ( [ ab]; , ( L 2 ) ) .Then f o r any t E [a,b], the white noise integral

1,"d , * ( g ( B ( s )B,( b ) ) )d s is a Hitsuda-Skorokhod integral and

Theorem 5.2. (Kubolg 1983) Let 8 E S'(R). Then f o r any 0 < a 5 b, the following equality holds in the space ( S ) *

where 9' and 8" are derivatives in the distribution sense. It6's formula can also be generalized to 9 ( X t ) with X , being a stochastic process given by a white noise forward integra125>30.But the conditions are rather complicated and technical. For the detail, see the papers by Asch and P o t t h o P ~and ~ ? ~by P ~ t t h o f f ~ ~ . 6. Stochastic integral equations

Let X a be measurable with respect to u{B(t)It 5 a } and f and g satisfy the Lipschitz and linear growth conditions. Then the stochastic integral 65

equation

x(t)= x a +

I’

f(s,~ ( s d) )~ ( s+)

s”

g(s, ~ ( s ds, ) )

(8)

has a unique continuous solution. Suppose the initial condition X a is not measurable with respect to a { B ( t )1 t 5 u } . Then we cannot use the iteration method to obtain the solution within the It8 theory. However, by using white noise theory we can rewrite the Equation (8) as

x(t)=xa+l a : f ( s , ~ ( s ) ) d s +

t

g(s,X(s))ds,

(9)

where the first integral is a Hitsuda-Skorokhod integral. In many cases we can use the S-transform to solve this equation as follows: (1) Take the S-transform of Equation (9) to get an ordinary differential =SX(t)(J). equation for Ft(J) (2) Solve the new equation for &(<) for each E SC(R). (3) Take the inverse S-transform of &(<) to get the solution X ( t ) .

<

We give several examples to illustrate this method and to show new and interesting phenomena beyond the It6 theory of stochastic integration.

Example 6.1. Buckdahn‘ considered the Skorokhod form of the following stochastic integral equation

~ ( t= )sgn(B(1)) +

1 t

a , * x ( s )ds,

o 5 t 5 1.

He used a rather complicated method to find the solution. However, it is very easy to use the S-transform method to derive the solution

~ ( t= )sgn(B(1) - t )eB(t)-it. Observe that the discontinuity of the sample paths of X ( t ) is due to the anticipative initial condition.

Example 6.2. Consider the stochastic integral equation

where t E [0, I]. We can use the S-transform method to derive the solution

~ ( = t )eB(t)-ft(l+ tsgn(B(1) - t ) ) . Observe that the discontinuity of the sample paths of X ( t ) is due to the anticipative integrand of the second integral. 66

Example 6.3. There are simple equations such as

X ( t )= 1 +

I’

a,*( B ( l ) X ( s ) ds, )

(10)

which cannot be directly solved by the S-transform method. This equation written in the Skorokhod form was considered by Buckdahn5. We can use the next theorem to derive the solution of Equation (10)

Theorem 6.1. Suppose f is a deterministic function in Loo[a,b]. Then the stochastic integral equation X ( t ) = 20

+

1 a,* t

( f ( s ) B ( b ) X ( s )ds, ) aI tI b,

a

has a unique solution in L2([a,b];(L2)) and is given by

x(t)= 20 exp [ B ( b )

Ju

t

f ( s )e-J.t

f ( ~ ) dd ~ ~

(s)

Example 6.4. Consider the stochastic differential equation25,29

X(t)= 1+

Jo”

a,*(B(l)o X ( S ) )ds, 0 5 t 5 1,

where o denotes the Wick product. The solution is given by

X ( t ) =-d

1

exp [-2(1+ 1t

+ t 2 ) (tB(l)Z-2(l+t)B(l)B(t)+B(t.

Example 6.5. Compare the stochastic integral equations

I’ + It +

X ( t ) = B(t)

B ( l ) X ( s )ds,

Y ( t )= B ( t )

B(1) o Y ( s )ds.

Their solutions are given by

l

) X ( t ) = B(t)+ ~ ( 1 ) e ( t - s ) B ( l ) B ( s cis,

y ( t )=

I

t

d,*e(t-S)B(1)-i(t-S)2ds.

67

7. Perspectives of white noise stochastic integration

We briefly describe several directions and topics for further research on white noise stochastic integration. (1) Hitsuda-Skorokhod integral We have the fact that u l < p ~ m L P ( S ' ( R ) ,c p ) (S)*. But how can we tell that a generalized function is a random variable belonging to LP(S'(R),p)for some p > l? It is very desirable to obtain a characterization theorem to answer this question. Such a theorem will be very useful to study the Hitsuda-Skorokhod integral and white noise stochastic differential equations. (2) Solutions of white noise stochastic integral equations Prove a regularity theorem to obtain a strong solution from a weak solution of a white noise stochastic integral equation X ( t ) = X, s," f(s,X ( s ) )ds, in particular, a stochastic integral equation of the Hitsuda-Skorokhod type X ( t ) = X , J,"d,*f(s,X ( s ) )ds.

+

+

(3) Extensions of ItB's formula

Itb's formula for O(B(t),B(b)), a 5 t 5 b, has been extended to O ( X ( t ) B(b)) , for a Wiener integral X ( t ) = J,"f(s)d B ( s ) in the book. How about when X ( t ) is a Hitsuda-Skorokhod integral? (4) Clark-Ocone formula The Clark-Ocone formula has been formulated and generalized in terms of white noise theory by de Faria-Oliveira-Streit", namely, 'p = E'p SRE(at IFt)dB(t) for any 'p E W1/2. This formula has been further generalized by Ngobi-Stan31 by using the second quantization operator I?( l(-m,tj).Aase-Bksendal-Privault-Ub~e have also generalized this formula. It would be useful to extend their results to the case when 'p is a generalized function in the space (S)" , in particular, when 'p is a random variable in the union Ul
+

(5) Girsanov theorem Recall the probability space (S'(R), p ) and the Brownian motion B ( t ) defined in Section 2. Let h ( t ) ,0 5 t 5 T , be a nonanticipating stochastic process such that E, exp :5 h(t)2d t ) < m. Define a

(i

68

probability measure Q on S’(R) by T

ds]

dP.

+

The Girsanov theorem states that the It6 process X ( t ) = B ( t ) h ( s )ds is a Brownian motion with respect to Q. It would be very interesting to generalize this theorem to the case when h(t) is not assumed to be nonanticipating. In this case, the It8 integral h(s)d B ( s ) needs to be replaced by a Hitsuda-Skorokhod integral. Such a formula will be useful to study the Black-Scholes theoory for anticipative markets.

Acknowledgments The author would like to thank the Academic Frontier in Science of Meijo University for financial supports and to give my deepest appreciation to Professors Y. Hara-Mimachi, T. Hida, K. Nishi, and K. Sait6 for the warm hospitality during my many visits to Meijo University.

References 1. Awe, K., Bksendal, B., Privault, N., and Ubae, J.: A white noise generalization of the Clark-Haussmann-Ocone theorem, with application to mathematical finance; Finance and Stochastics 4 (2000) 465-496 2. Asch, J. and Potthoff, J.: A generalization of It& lemma; Proc. Japan Acad. 63 (1987) 289-291 3. Asch, J. and Potthoff, J.: White noise and stochastic analysis; in: Stochastics, Algebra and Analysis in Classical and Quantum Dynamics, S . Albeverio et al. (eds.), Kluwer Academic Publishers (1990) 4. Asch, J. and Potthoff, J.: It6’s lemma without non-anticipatory conditions; Probab. Th. Rel. Fields 88 (1991) 17-46 5. Buckdahn, R.: A linear stochastic differential equation with Skorohod integral; Markov processes and control theory, Math. Res. 54 (1988) 9-15 6. Buckdahn, R.: Anticipating linear stochastic differential equations; Lecture Notes in Control and Information Sciences 136 (1989) 18-23, SpringerVerlag 7. Buckdahn, R.: Anticipative Girsanov transformations; Probab. Theory Related Fields 89 (1991) 211-238 8. Buckdahn, R.: Skorohod stochastic differential equations of diffusion type; Probab. Theory Related Fields 93 (1992) 297-323 9. Buckdahn, R.: Anticipative Girsanov transformations and Skorohod stochastic differential equations; Mem. Amer. Math. SOC.111 (1994) no. 533 10. Buckdahn, R. and Nualart D.: Skorohod stochastic differential equations with boundary conditions; Stochastic Stochastic Rep. 45 (1993) 211-235 69

11. de Faria, M., Oliveira, M. J., and Streit, L.: A generalized Clark-Ocone formula; Random Oper. Stoch. Eqs. 8 (2000) 163-174 12. Hida, T.: Andpis of Brownian Functimals. Carleton Mathematical Lecture Notes 13, 1975 13. Hida, T., Kuo, H.-H., Potthoff, J., and Streit, L.: White Noise: A n Infinite Dimensional Calculus. Kluwer Academic Publishers, 1993 14. Hitsuda, M.: Formula for Brownian partial derivatives; Second Japan- USSR Symp. Probab. Th. 2 (1972) 111-114 15. Hitsuda, M.: Formula for Brownian partial derivatives; Publ. Fac. Integrated Arts and Sciences, Hiroshima University, Series III 4 (1978) 1-15 16. Holden, H., 0ksenda1, B., Ubpre, J., and Zhang, T.: Stochastic Partial Dijferential Equations. Birkhauser, 1996 17. Its, K.: Stochastic integral; Proc. Imp. Acad. Tokyo 20 (1944) 519-524 18. It6, K.: Extension of stochastic integrals; Proc. Intern. Symp. Stochastic Differential Equations, K. It6 (ed.) (1978) 95-109, Kinokuniya 19. Kubo, I.: It6 formula for generalized Brownian functionals; Lecture Notes in Control and Information Scis. 49 (1983) 156-166, Springer-Verlag 20. Kubo, I. and Kuo, H.-H.: Finite dimensional Hida distributions; J. Funct. Anal. 128 (1995) 1-47 21. Kubo, I. and Takenaka, S.: Calculus on Gaussian white noise I; Proc. Japan Acad. 56A (1980) 376-380 22. Kubo, I. and Takenaka, S.: Calculus on Gaussian white noise 11; Proc. Japan Acad. 5 6 A (1980) 411-416 23. Kubo, I. and Takenaka, S.: Calculus on Gaussian white noise 111; Proc. Japan Acad. 57A (1981) 433-437 24. Kubo, I. and Takenaka, S.: Calculus on Gaussian white noise IV; Proc. Japan Acad. 5 8 A (1982) 186-189 25. Kuo, H.-H.: White Noise Distribution Theory. CRC Press, 1996 26. Kuo, H.-H.: Anticipatory It& formula and Hitsuda-Skorokhod integral; in Skorokhod’s Ideas in Probability Theory, V. Korolyuk et al. (eds.), Proc. Institute of Math., National Academy of Sciences of Ukraine, Vol. 32 (2000) 248-255 27. Kuo, H.-H.: White noise theory; in Handbook of Stochastic Analysis and Applications, D. Kannan and V. Lakshmikantham (eds.) 107-158, Marcel Dekker, 2002 28, KUO,H.-H.: A quarter century of white noise theory; in: Quantum Injormation IV, T. Hida and K. Sait6 (eds.) (2002) 1-37, World Scientific 29. Kuo, H.-H. and Potthoff, J.: Anticipating stochastic integrals and stochastic differential equations; White Noise Analysis-Mathematics and Applications, T. Hida et al. (eds.) (1990) 256-273, World Scientific 30. Kuo, H.-H. and Russek, A.: White noise approach to stochastic integration; J . Multivariate Analysis 24 (1988) 218-236 31. Ngobi, S. and Stan, A.: An extension of the Clark-Ocone formula; International J. Math and Math Sciences 1463-1476 (2004) 32. Obata, N.: White Noise Calculus and Fock Space. Lecture Notes in Math. 1577, Springer-Verlag, 1994 70

33. Potthoff, J.: Stochastic integration in Hida’s white noise analysis; in: Stochastic Processes, Physics and Geometry, S . Albeverio et al. (eds.), World Scientific (1990) 34. Potthoff, J. and Streit, L.: A characterization of Hida distributions; J. Funct. Anal. 101 (1991) 212-229 35. SaitB, K.: Its’s formula and Lkvy’s Laplacian; Nagoya Math. J. 108 (1987) 67-76 36. SaitB, K.: ItB’s formula and Lkvy’s Laplacian 11; Nagoya Math. J. 123 (1991) 153-169 37. Skorokhod, A. V.: On a generalization of a stochastic integral; Theory Probab. Appl. 20 (1975) 219-233

71

CONNES-HIDA CALCULUS A N D BISMUT-QUILLEN S U P E R C O N N E C T I O N S

R. LQandre Institut dc Mathbmatiques. Universitk de Bourgogne 21000. Dijon. FRANCE email:Flemi.leandreOu-bourgogne.fr H. Ouerdiane Dkpartement de Mathbmatiques. Facultk des Sciences de Tunis 1060.Tunis. TUNISIA email:[email protected] Abstract: We show that the J.L.O. cocycle for the Index theorem for a family of Dirac operators, by using Bismut-Quillen superconnection formalism, is a white noise distribution. Mathematics subject classification (2000): 58520. 58B34. 58J65. 60H40. K e y words: J.L.O. cocycle. White noise analysis. Bismut-Quillen Superconnection. I. Introduction Let us remark that there are 3 infinite dimensional distribution theories: -1) White noise analysis ([B.K], [B], [HI, [H.K.P.S], [St]), which uses the Fock space and which is

algebraic. -2)Watanabe’s distribution theory [Wall, which is a part of Malliavin Calculus ([Mall), and which is probabilistic. -3)Non-commutative differential geometry of Connes ([Coz]) which is algebraic. The goal of this paper is to show that the first one is very similar to the third one, if we disregard the operations (the algebra of annihilation and creation operators for white noise analysis and the cyclic complex for non-commutative differential geometry) for the particular case of the algebra of functions on a manifold. The auxiliary bundle associated to a twisted Dirac operator plays an important role in the Index theorem. Let us recall that any complex bundle [ on a compact manifold endowed with a connection V is isomorphic to a subbundle of a trivial complex bundle Cd endowed with the projection connection. So [ is given by p ( C d )where p is a smooth complex projector. If we consider the Dirac operator D which acts on the sections of the spin bundle S of a spin manifold, the twisted Dirac operator is p D p which acts on the section of S @ [ . There are two ways to study the algebraic properties related to it: -l)Bismut ([Bill) introduced its Chern Character over the free loop space, showed it is related to the equivariant complex on the free loop space. A beginning of rigorous interpretation in Watanabe’s sense of the works of Bismut is given in [J.L], [Ld], [Ls], [Le],[LT],[Lg] which give a probabilistic interpretation of the considerations of [G.J.P]. A beginning of interpretation in the white noise sense of the works of Bismut is given in [Ls] and [Llo] starting from the work of Getzler ([GI). -2)The introduction of the cyclic complex and the Chern character in cyclic homology ((CO~], [G.S], (J.L.01, [J])explains too the algebraic properties of the auxiliary bundle. These considerations are a part of non-commutative differential geometry. In -1) and -2), the authors try to give a meaning to the Index theorem for a twisted Dirac operator as a current, which operates in -1) on forms on loop space and in -2) on the cyclic complex of the algebra of smooth complex functions on the manifold. In [Ls] and in [L~o], we have shown that the symmetric Fock space plays a big role in order to understand Witten’s current, by using Chen forms. There are others Fock spaces than bosonic Fock space and Fermionic Fock space. It is for instance the case of the interacting Fock space of Accardi-Boaejko ([A.B]]. We consider the interacting Fock space of Accardi-Bozejko in order

72

to understand the natural criterium of convergence of series which appears in entire cyclic cohomology of Connes ([Col]). We recover the algebra of smooth functions on a compact manifold as the intersection of some Hilbert-Sobolev spaces on the manifold, by using the Sobolev imbedding theorem. We are concerned in this paper by the algebraic properties associated to the Index theorem for families, by using Bismut-Quillen superconnection formalism. In the case of non commutative differential geometry, it is studied by [Nil, [N.T] and [PI,We are motivated to interpret the previous considerations in the framework of white noise analysis. In the case where we use Bismut-Chern character, it is the purpose of [Llo]. The main novelty of this work is the following: M a i n theorem:the J.L.O. cocycle for a family of Dirac operators is a white noise distribution with values in the space of smooth forms on the basis manifold. We use probabilistic technics in order to estimate densities. The reader interested by various probabilistic proofs of the Index theorem, including Bismut's proofs can see the surveys of LQandre[Lz], Kusuoka ([Ku]) and Watanabe ([Waz])about that topic. The reader interested by probabilistic proofs of the Index theorem for families, besides the original paper of Bismut ([Biz]),can see the pedagogical paper of LBandre ([Ll]) with a simplified proof. The second author thanks the warm hospitality of the Institut Elie Cartan where this work was done. 11. Cyclic homology and white noise analysis

Let us consider a compact Riemannian manifold M of dimension d endowed with the Riemannian measure d m ~Let . A be the Laplace-Beltrami operator over it. A = dzvgrad where gradf(z) is the vector associated to the 1-form d f ( 2 ) via the Riemannian structure. A is self-adjoint, densely defined over the Hilbert space HO= L 2 ( M ) of complex functions on M . We consider the Sobolev space Hk (k E N ) of complex functions 4 on M such that /,(A l ) k 4 & h ~< M. The Laplace-Beltrami operator A has eigenvalues A, 2 0 (numbered in increasing order) associated to the eigenfunctions 4., Only A0 = 0 and XO is associated to 1. Moreover, An Cn' when n -+ co for some integer T ([Gi]). Let k E N , C > 0. We consider the space Wk,C constituted of elements F

-

+

where F,, belongs to Hk equipEed with its natural Hilbert structure. f f k = Hk/C, and we assimilate an element $ of Hk to its image in Hk (We work on the complex numbers). Moreover, Hk+l c Hk. We define over Wk,c the following Hilbert structure:

called an interacting Fock space ( [A.B], [A.N], [G.H.O.R], [Ou], [O]). Definition 11.1: The space of white noise test functionals is &CN,C>OWk,C= W endowed with the projective topology. (See [HI). W is a nuclear Frechet space, because the Sobolev imbedding theorem. We consider the Hochschild boundary (See [Lo] for instance) Wk,C is

n-1

(2.3)

b(4o 8 41 8 .. 8 4,) = x ( - 1 ) ' 4 0 8 .. 8 da4i+i8 .. 8 4n

+ (-1)"&4o

8 41.. 8 4"-i

i=O

T h e o r e m 11.2: The Hochschild boundary b is a continuous linear application from W into W . Proof: The proof is very similar to the proof of ThBoreme 11.5 of [Ld]. Let us write: (2.4)

73

where I = (io,._,in) and 111 = n. We have (2.5) We use ( 2 . 3 ) and Sobolev imbedding theorem (See [Gi]) in order to find a kl > k and CI

We consider some big kz

> C such that

> kl and some small CZsuch that

(2.7)

if kz is big enough. We deduce by Cauchy-Schwartz inequality that for some big C3 (2.9)

IlbF/lk3C

>0

5 AIIF11Zki,C3

Therefore the result.

0 We consider the Connes operator ([Coz], [Lo]) n

B(4o @ 41 @ .. @ 4n) = c(-1)" 41 i . .@ @ 4 n @ 40 @ 41..@ 4 i - I

(2.10)

i=O

We have the following theorem, whose proof is omitted because it is similar to the proof of Theorem 11.2: Theorem II.S.:B is a linear continuous map from W into W . As classical (See [Lo], [COZ]for instance), we have the relations b2 = B2 = bBf Bb = 0

(2.11)

+

such that b B is a complex. Definition 11.4.: b + B operating on W is called the cyclic complex in white noise sense. Let us give some classical examples of elements belonging to the cyclic complex (See [G.S]). We denote

Hm- = nk>oHk.

(2.12)

By the Sobolev imbedding theorem, Hm- is nothing else that the algebra of smooth functions on the manifold. Let p = ( p i , ) ) belonging to some Mn(Hm-) ( the space of ( n x n) matrices with component in H m - ) such that p 2 = p. It defines a complex bundle over M . Reciprocally, each complex bundle ( endowed with a connection V can be given by some projector belonging to some M,,(Hm-) endowed with the projection connection (See [N.R]). We denote (See [GS] p 346): (2.13)

p n = Tr@@.. @ P ) =

C

p i o , i , @ P i l . i z . .@ p i - - , , i , , @ P i " , i o

;a,. ..in

74

and

(2.15)

since p 2 = p and since the image of 1 in pk is equal to 0. We denote (See [GS] Proposition 1.1) (2.16) k=l

It is clear that Ch,(p) projector belongs to the space W of white noise test functionals for every complex projector p and that (2.17)

( b + B)Ch*(p)= 0

(See [G.S] Proposition 1.1 for the proof of this identity which comes from (2.15)). In order to show that Ch,(p) belongs to W , we remark that:

and that (2.19)

111. T h e J.L.O. cocycle as a w h i t e noise distribution

-

Let us consider a fibration ?r : M + B of compact Riemannian manifolds:a-'(O) 0 x V for some open neighborhood of each point y in B . The generic element of B is denoted by y and 2 denotes the generic element of the fiber V,. We denote by A ( B ) the exterior bundle on B: we consider ?r*(A(B))the pullback bundle on B endowed with the trivial connection V. We suppose that the fiber V, is spin, and we consider the spin bundle S, = S,' CB S; over V,. Let us recall what we mean by that. We consider the double cover Spin(d) of SO(d), where the dimension d of V, is supposed even. We consider SO(V,) the frame bundle of V, supposed ocientable. We would like to get a lift of the principal SO(d) bundle SO(V,) by Spin(d), called Spin(V,). It is not always possible. We have to suppose that some topological constructions are satisfied. Moreover, we have the spin representation of Spin(d) on S+ CB S-. We consider the associated bundle S,' @I S; on V, supposed spin. The construction of Spin(V,) is not canonical, and supposed some choices. We suppose that the bundles S, fit together in a complex bundle S on M . Let us recall that the Clifford algebra of the Euclidean space Rd endowed with its canonical Euclidean structure <, . > is the algebra constructed from Rd where we have the relation for e and e' in Rd: (3.1)

e.e'

+ e'.e = 2 < e, e'

75

>

.

Since the construction of the Clifford algebra on Rd does not involve any choice, there is no problem to construct the Clifford bundle CI, on V,. We assimilate the Clifford bundle with its complexification. The Clifford bundle CI, acts over S,, and the product by an element of the tangent bundle, considered as a subbundle of the Clifford bundle, is odd relatively to the natural graduation on S,. We consider the LeviCivita connection V' on V,, which passes to S, because Spin(V,) is a lift of SO(V,) by Z/2Z and to CI,. We consider the family of Dirac operators D,. In local coordinate, D, = C e;,,V&, where ei,, is a local orthonormal basis of T(V,). We consider the Levi-Civita connection V M on M . According Bismut ([Biz]), we introduce another connection V' on M . We have V',Y = IIVxY if Y is a vector field on V,, where II is the orthonormal projection from the tangent bundle of M on the tangent bundle on Vu. In particular, if X is a vector field on Vu,V>Y = V g Y . If Y is the pullback of a vector field on B,V',Y = 0 if Y is a vector field on V,. Moreover Vk.xr*Y is the pullback of V$Y where VB is the Levi-Civita connection on B. In order to do these considerations, we have chosen the metric on M such that the orthogonal bundle of V, is isometric to the tangent space of B by the derivative of r. We get by using this orthonormal decomposition of the tangent bundle of M into the tangent bundle of V, and its orthogonal the notion of pullback of a vector field on B in a vector field on M . We have, since connections differ by one form VM = V'+

(3.2)

s

where S is a one form with values in the tensorial operator on T ( M ) . Let E, be the bundle on V, r * A ( B ) 6 S y .Over Xu,we consider the connection V v = 0' coordinate, it is equal to

+ S. In local

where e, denotes a local orthonormal basis of T(V,) which acts by Clifford multiplication on S, and f, an orthonormal basis of T ( B )which acts by exterior multiplication of A ( B ) (See [Ll](3.41)). Let us consider Hm = H+" fBH_" the infinite dimensional bundle on B of smooth sections of S,. Let us define a trivialization of Hm. Let 0 be a small open ball centered in yo in B. There is a unique geodesic joining yo to y in 0 called IB(YO,Y). Since r * T ( B )is supposed orthogonal to T(V,)in T ( M ) ,we can lift this curve in a curve lyo,,(z).$0 -+ I,,,,,(~o) realizes a diffeomorphism from V,, to V, if y is close enough from yo. The connection V' preserves the orthogonal decomposition of T ( M ) intp T(V,)and r * T ( B ) . Therefore, the parallel transport along lye,, realizes an isomorphism between H E and H F which preserves the Z/ZZ graduation. We consider the bundle on B A ( B ) 6 H m . We consider a Z/ZZ graded tensor product. r*Au(B)6CIuacts on r*A,(B)6H,m, but we have to take care that we consider 2 / 2 2 tensor products. We have namely: (3.4)

(u6e;)(u6$,)

= ( - l ) l r l d e g o u A u16ey$'

if ey = eh..e;" where n = 111. If u is a smooth form on B and $Ju a section of S,, we write: (3.5)

D y ( u 6 $ ~=) (-l)deg"u6Du$JY

Let
ds

+ Cdf,,A,,

= Vs

where a , = (cY,~, ..,a,,) and dfo, == df,., A .. A df,., (T = larl). If IarI is even, df,, is an even operator such that A,, is an odd operator acting on the local sections on < B which do not involve derivatives of these

76

sections. If larl is odd, df,, is an odd operator (which acts by wedge multiplication according the sign rules (3.4)) such that A,, is an even operator acting on the local sections on E E , which acts without to take some derivatives of these sections. We put if A,, is even:

where A,, is given in diagonal form as (A&,A;,). If A,, is odd trsA,, = 0. The curvature R, = 0: is a tensorial operator, which is even, and which operates on A(T(B))6,EE (we have to take care with conventions of signs analog to (3.4) in order to do the product). The big difference with the traditional theory, where the curvature of a connection of a bundle is a 2-form, is that we can consider forms of any degrees in R.. Bismut ([Biz]) has extended this formalism to the case where we replace EB by Hm. If a1 = I, A8 = D, and if ( ( X I ( 2 1, A,, is an odd operator which is in some sense tensorial: it acts fibers by fibers on *'A(T(B))6Su without to take derivatives of the section of this bundle. In order to do the asymptotics Bismut used a special superconnection. We follow him. We consider the Bismut superconnection, called by Bismut the Levi-Civita superconnection: (3.8) (See [Biz], [Lil). We consider over the space of operators on A(TB)&IH" the natural 2 / 2 2 graduation, such that V r is an odd operator. We can consider the curvature R? of 0:. It is an operator which is tensorial in y and a differential operator in z whose part of second order is ( D u ) 2 . Let us consider a unitary complex bundle E, on V, endowed with some unitary connection Vc*. We can define the associated horizontal Laplacian acting on sections of this bundle. Let V' be the Levi-Civita connection on V, and ey be a local orthonormal basis of T(V,).Let & be a section of E Y . We have: (3.9)

AV'" is a densely defined self-adjoint operator acting on the La sections of E,., We will consider in the sequel the special case of the bundle r'A(T(BG))&IS,endowed with the connection 9,. Let K"z) be the scalar curvature on V, for the Riemannian metric on V,. We get the Lichnerowicz formula (See [B,], [Ll] Theoreme 111.1): (3.10)

RTy = &f+ 1/4K'

where A: is the horizontal Laplacian -TT(@')~ which acts on the section of the bundle E , on V,. Let zr(z) be the Brownian motion on V, starting from z. It is the solution of the stochastic differential equation in the Stratonovitch sense (See [El], [Em], [I.W]) (3.11)

dZ:(Z)

= TF(Z)d&

.

where Bt is a flat Brownian motion in Tz(V,), T ? ( Z ) the stochastic parallel transport on V, along the Brownian path for the Levi-Civita connection on V,. We can use another way by introducing the bundle of direct orthonormal frames SO(%) and the canonical vector fields X:(z) on it. Let us recall what are these objects. The frame bundle SO(V,)is the space of oriented isornetries u u from the canonical Euclidean space Rd into T(V,(z)).It inherits from the Riemannian structure the Levi-Civita connection. Let e, he the canonical oriented orthonormal basis from Rd. u Y ( e , ) constitutes a direct orthonormal basis of T(V,(z)). We consider the horizontal lifts X:(u) of uY(e,). The set of X:(u) constitutes the set of canonical vector fields of SO(V,). Let A, the projection from O(V,) on V, and let us consider the following Stratonovitch equation (See [LW]) starting from u: (3.12)

77

with Bf a fixed Brownian motion in Rd. We have a representation of the Brownian motion in z:(z)

=

+(.l(.)).

<,

Let be the complex bundle on V, considered as before. Let T ~ ( zbe ) the parallel transport along the Brownian path t + zy(z) for the bundle E, endowed with the connection Vcu. We have the following stochastic representation to the semi-group associated to A"'" : exp[-t/2Av'"]@(z)

(3.13)

.

= E[(TjY (z))-'$b (zy(z))]

We deduce a stochastic realization of the semi-group associated to Rr, by using the Feynmann-Kac formula. The semi-group exp[-t/2RFu] acting on smooth sections p' of Ey is given by the following probabilistic representation (See [Biz] [LI]):

where i / ( z ) is the stochastic parallel transport along the Brownian path t -t z:(z) Let F belonging to W . Let us introduce the J.L.O. cocycle Ch*(Vy)by

for the connection

VY.

where [.,I denotes the supercommutator. Let us recall that if A and B are two operators acting on the sections of E considered as bundle on M ,

[ A , B ]= AB - (-l)degAdegBBA

(3.15)

as the usual supercommutator. Since Q is even,

IVY, 41 = V F 4 - 4vr

(3.16)

.

Let us denote by dzq5 = xe:(z)&Q(z,) in a local normal coordinate system 2, of V,. Let us denote by dub = C fe&4 in a local orthonormal basis of B after trivializing the fibration M + B. In some sense we We have have identified A ( T ( M ) )to x*A(T(B))6CL(V). P?>Ql=dd+dg4.

(3.17)

Unlike the traditional J.L.O. cocycle for a single operator, our J.L.0 cocycle takes its values in the form on B. Let us recall in order to understand the definition that the supertrace Tr, of an operator transforming h,(B)&,+(z) into A,(B)&S$(z) is its partial trace in S $ ( x ) . If the operator interchanges S+ and S-, its supertrace is equal to 0. If we consider an operator from AV(B)6S;(z) into himself, its supertrace is the opposite of its partial trace in S;(z). Let us recall that the supertrace of a supercommutator is equal to 0 ([Qil)

Moreover, exp[-tR$' ]'

has an heat kernel qy(z,z'). We have:

4; exp[-slR?][VF, Q;] exp(-(sz

$3exp[-(l-

- sdRF]..[VY,

(3.18)

where dmv, is the Riemannian measure on V,

78

sdR,"Id~~(z)

Therefore, T ~ . { & e x ~ [ - s i R r ] [ V rSil , exp[-(sz - sdRr]...[v?, (3.19)

=

s,-+>

(&4: Over A ( B ) , we consider the Laplacian d L d s spaces: (3.20)

+ dy4:)(zn)qY-s,(Zn,

+ dkds = A,

~ ~ d ~ ( ,= q , h

(Ak, f

41:

exp[-(l-

~n)Rrll

+~'43bl)...

T..{~x(.)4:,(z,zl)(dz4i

.

Z)}~~V,(Z)...~~V"(Z,)

and for k belonging to N , the various Soholev

l)U,O

> dmB

Ch norms of u can be estimated by the systems of norms (3.20) by Sobolev imbedding theorem ([Gi]). We have: Lemma III.1:Let HYl,.,,sn($o, ..,+") (SI < sz < .. < s, < 1) be the operator acting on the sections of E,: (3.21)

$'

+

4'exp[-siRT']4'..4"exp[-(l

- s,)R~,]$J'

where the 4, are tensorial operators acting on section of the bundle E , considered as a bundle on M . Then (3.22)

H:,,.+"

..I

4")3'(4 =

L

q:l,.,,sn(40, ..,P ) ( z ,z')$JYz')dmv,(z')

.

Moreover the covariant derivatives in y, z and z' of the kernel qf,,,,,s,,($o, ..,@"'(z, z') can be estimated by C " n 11q411k for an integer k which depends only from the order of the derivative. Proof: We consider a probabilistic representation of HY,,,,,sn($o, ..,@").It is given by

where ?#,:, (z) if s < s' is the stochastic parallel transport for the bundle E, along the Brownian path z:(z) runned in the opposite sense from zy,(z) to z:(z). We neglect the fact in (3.23) that the Brownian motion represents the semi-group associated to RTv/2 and not the semi-group associated to R?'. We would like to apply Malliavin Calculus to estimate the density of HY,,,,,sJ@"'..,@'). Let H be the Hilbert space of functions h from [0,1] into Rd such that h(0) = 0 and such that Id/dsh(s)12ds< w. It is the reproducing Hilbert space of the Brownian motion t + B;. In order to define the Sobolev norms of a Wiener functional F , we take its derivatives in the direction of H . Its derivative of order T is realized as a random element V'(s1, ..,s r ) of the symmetric rth-tensor product of H . The Sobolev norms of Malliavin Calculus are given by:

Ji

We work in a trivialization 0 x V of the fibration M . &ur(z) belongs to all the Sobolev spaces of Malliavin Calculus, after imbedding V in a l i e a r space R" and trivializing SO(V,) by imbedding sod into Gl(R").The derivatives in z, y of the functionals which are considered are bounded in all the Sobolev spaces of Malliavin Calculus by Cnll@llh for some k (We use Sobolev imbedding theorem). The same results holds namely for &?Y,,82(z) after trivializing the family of bundle Ey considered as a bundle on M . Moreover, the Malliavin matrix of z:(z) is uniformly bounded in all the L p in z and y. The results holds by Malliavin

79

Calculus (See [Nu], [I.W], [L1]p 394.). Namely for any vector fields X I ,..,X , on M in a local trivialization 0 x V of the fibration M , we have:

where

11.1103

denotes the supremum norm.

0 Remark Instead of using the non intrinsic Malliavin Calculus, we could use the geometrical Malliavin Calculus on a manifold developed by Bismut and LQandrein order to state this lemma (See for instance [L5] and [L6]). Lemma 111.1 allows us to state the following theorem: Theorem 111.2: The J.L.O. cocycle Ch*(Vy) is a white noise distribution (an element of the topological dual W' of W) with values in L I ( B ) ~ - . Proof: We write: (3.26) Then

(3.28) for some k' by Lemma 111.1. But, on the other hand,

(3.29)

Il4*>Ilk,

= (A,,

+ 1)"'*

,

By proceeding as in Theorem 111.2, we deduce that there exists kl and Cl independent of F such that (3.30)

II < Ch*(V?),F > IIA(B),~5 CIIFllk,,cl

0

Lemma III.3:Let q:(x, z') be the heat kernel associated to the heat-semi group associated to RFu and V and V' first order operators acting on E,. Then (3.31)

(3.33) where d$(., .) denotes the Riemannian distance on V, and where t 5 1

80

Proof: We follow the method of the proof of [La] of this fact for the scalar heat kernel. Let h be a mollifier function equal to zero outside a small convex neighborhood of z. Then the kernel associated to the operator

is bounded as well as its derivatives by exp[-C/t] by using the tools of Malliavin Calculus, because by exponential inequality P{d,(z,z:z)) > 6 > 0) 5 exp[-C/t] and because the inverse of the Malliavin matrix associated to z:(z) is bounded in LP by Ct-n(P) when t + 0. So it is enough to study the density of

(3.35)

+

~y(~~(~))~~/~l$y(~~(~))l .

E[h(zf(z))$,&)

We do the traditional time scaling, in order to come back at time 1 by replacing dB; by t1/2dBBin order to replace the short time asymptotic by a the study of a diffusion in time 1 which depends from a small parameter (See [Mo]). But we don’t change the notation, in order to me more succinct. On the other hand, there is a C such that e~p[C~;(~’;‘(’))] as soon as z f ( z ) is close enough of has Sobolev norms in Lz of each order as its derivatives in z and y bounded (See [La]). Namely, we have the following large deviations estimates (See [F.W], [L3]) when t + 0:

4

where Int 0 denotes the interior of the borelian subset 0 of V, and clos 0 its closure. We consider the operator p: (z):

&

It has a kernel by proceeding as in [Lz] bounded by with first derivatives bounded by &. We operate as in [La] to do that. We work in normal coordinates around z in V,. We consider the rescaled operator ur (z):

with some natural notations, because E, is locally trivial around z. It has a bounded density as well as its derivatives when t + 0. Let us denote by this density. The supremum norm of the density of p:(z) is bounded by t-d/zsup,, lQr(z, Analogously statements works for the derivatives of the density of P:(x). We conclude as in [La].

0. This lemma allows to show: T h e o r e m 111.4:(b + B)Ch*(VY)= 0. Rem ar kThi s means that < Ch’(VY),( b B ) F >= 0 for all F belonging to W . ProoEThe proof is exactly the same than in the Proof of Theorem A of [G.S], the bound of Lemma 111.3 allowing to justify the algebraic computations. It is enough to apply these bounds, and the Kolmogorov relation in the classical bound of the heat kernel p:(z, associated to the Laplace-Beltrami operator on V, as in [J.L]:

+

(3.39) for t 5 1

81

0 Let us consider for

T

E [0,1] the superconnection VF7:

(3.40)

vTT=v:+'d2P-1)1v?,P]

(See [G.S] p 357.). It has a curvature RF7. Lemma 111.5: The heat kernel associated to RFT satisfies the same estimates than the heat kernel associated to R F in Lemma 111.3. Proof: Let us write RFT = RF + A where A is a first order operator. We apply the Volterra expansion:

(3.41) exp[-tR:]

+

(-1)"exp[-siR~]Aexp[-(sz - s~)R:]A..Aexp[-(t

- s,)R;]dsI..d~,

.

By using (3.38), Lemma 111.3 and Kolmogorov formula, the kernel q?(z, z') of I , satisfies to (3.42)

14;(%

z')l

1

5 P%A%.') l
dsl..dsn

J s l ( s z - sl)..(t - Sn)

.

But (See [J.L] p 144.)

1

(3.43) %./SI(SZ

- S l ) . . ( t - Sn)

dsi..ds,

C"

5-

a.

The result arises when we apply (3.39) to p & ( z , z'). We estimate the first derivatives of the kernel of I,, by using the same arguments.

0 We can d e h e Ch*(VF7) by using Lemma 111.4 and show that it is a Hida distribution, if we disregard the smoothness. The estimates of the first derivatives of the heat kernel associated to R G allow to justify the algebraic considerations of [GS] and to show that: Lemma III.fkd/dT(< Ch'(VTT),Ch.(p) 2)= 0. We deduce that: Let us state the main lemma, analogously to Lemma 2.2. of [GS] in order to show Theorem 111.4 and Lemma 111.6. We consider some tensorial operators @ acting on sections of E. considered as a bundle on M . We consider the expression: (3.44)

< H'($O, ..,$") >=

J

Tr, { doexp[-sl RT7]$'.. .$" exp[- (1 - s,)RTT]}dsl. .ds,

O
The difference with the delinition of [GS] p 348, is that < H'($O, ..,@') > takes its values in the space on forms on B unlike the scalars. We remark first of all that: -i)[VFT,$] is a first order operator. -ii)[RFT,$]is the sum of composition of first order operators. -iii)[VcT,d / d ~ V ? ~is] a first order operator. These remarks and the bounds given in Lemma 111.5 allow us to state the analog of Lemma 2.2. of [GS] with values in forms unlike scalars, because the supetrace of a supercommutator is equal to 0: Lemma 111.7: Let us consider some tensorial operators @ acting on the sections of E. considered as a bundle on M . We have: (3.45)

<~

~ ( 6 , ,0, p, n ) >= ( - l ) ( d e s i ' + . . . + d e s e ' - ' ) ( d ~ s e ' + . . + d . s d " )

82

< H'(d',

..$",$O,

..,$$-I)

>

(3.48)

(3.49)

< H'(&, .., [ R G ,41,,.,6") >=< H'(cj0, ..,@-'#,..,$") > - < HT(40,__, @-',@f+',..qY"' dldr

< H'(d0, ..,6") > +

n

C < H'($',

>

.., @, [VrT,d/drVTT],@+',..,qbn) > .

*=O

Let us recall that KerpD,p can get some jumps of dimension when y is moving. CokerpD,p can get too some jumps of dimension. This means that y + KerpD,p does not in general define a bundle on B. But since the kernel of pD,p and its cokernel have the same jumps of dimension, this justifies that 1ndpD.p = Ke7pD.p - C0kerpD.p is a virtual bundle in complex K-theory sense. It is the virtual Index bundle of Atiyah and Singer associated to the family of twisted Dirac operator pD,p. If = Ch(lndpD.p) where Ind(pD p) is the Index bundle on B of the family of twisted Dirac operators y + pD,p. Proof: We remark that if T = 1, it is nothing else than the theorem of Bismut ([Biz],[Ll])

< Ch*(VFl),Ch,(p) >= Ch(lndpD.p)

(3.51)

where Ch is the Chern character in real phase of the Index bundle, because in such case p commute with VTl such that (See [GS] p 356.) (3.52)

< Ch*(Vrl),Ch,(p) >= Tr,pexp[-Rr1]

= Tr,pexp[-Rr1]p = Ch(1npD.p).

The result arises by Lemma 111.6.

0 IV. References [A.B] Accardi L. Bozejko M.: Interacting Fock spaces and gaussianization of probability measures. Inf. Dim. Ana. Quant. Probab. Rel. Top 1 (1998), 663-670. [A. N] Accardi L. Nahmi M.: Interacting Fock spaces and Orthogonal Polynomials in Several Variables. In "Non-commutativity, Infinite-Dimensionality and Probability at the Crossroads". Obata N., Matsui T., Hora A. Edt. World Scientific (2003), 192-205. [At] Atiyah M.: Circular symmetry and stationary phase approximation. In "Colloque en I'honneur de L. Schwarta" Asterisque 131 (1985), 43-59. B] Bereaansky Y.M.: Infinite dimensional non-Gaussian analysis connected with generalized translation 0perators.h "Analysis on infinite-dimensional Lie groups and algebras". H. Heyer J. Marion edt. World Scientific (1998), 22-46. [B.K] Berezansky Y.M., Kondratiev Y.O.: Spectral methods in infinitedimensional Analysis. Vols 1, 2, Kluwer (1995).

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[B.T] Berezansky Y.M. Tesko V.A.: the spaces of test and generalized functions, connected with generalized translation. Ukrainian Math. J 55 (2003), 1587-1657. [B.V] Berline N. Vergne M.: A proof of Bismut local index theorem for a family of Dirac operators. Topology 26 (1987),435-463. [Bill Bismut J.M.: Index theorem and equivariant cohomology on the loop space. Com. Math. Phys. 98 (1985), 213-237. [Biz] Bismut J.M.: The Atiyah-Singer for families of Dirac operators: Two heat equations proofs. Invent. Math. 63 (1986), 91-151. [Bi3]BismutJ.M.: Localization formulas, superconnections and the Index theorem for families. Com. Math. Phys. 103 (1986), 127-166. [C.K.S] Cochran W.G., Kno H.H., Sengupta A,: A new class of white noise generalized functions. Inf. Dim. Ana. Quant. Probab. rel. Top. 1 (1998), 43-67. [Col] Connes A.: Entire Cyclic cohomology of Banach algebras and characters of 0-summable Fredholm modules. K-Theory 1 (1988), 519-548. [Coz] Connes A.: Non-commutative differential geometry. Academic Press (1994). [Do] Dowelly H.: Local index for families. Michi. J. Maths. 35 (1988), 11-20. [El] Elworthy K.D.: Stochastic differential equations on manifolds. L.M.S. Lectures Notes Series 20. Cambridge University Press (1982). [Em] Emery M.: Stochastic Calculus in Manifolds. Springer (1989). [F.W]Freidli M. Wentzell A,: Random perturbations of dynamical systems. Springer (1984). [G.H.O.R.] Gannoun R., Hachaichi R., Ouerdiane H. Rezgui A.: Un theoreme de dualit6 entre espaces de fonctions holomorphes a croissance exponentielle. J. h n c t . Ana. 171 (2000), 1-14. [GI Getzler E.: Cyclic homology and the path integral of the Dirac operator. Preprint (1988). [G.J.P] Getzler E. Jones J.D.S. Petrack S.: Differential forms on a loop space and the cyclic bar complex. Topology 30 (1991), 339-371. [G.S] Getzler A. Szenes A,: On the Chern character of a theta-summable Fredholm module. J. h n c t . h a . 84 (1989), 343-357. [Gi] Gilkey P.: Invariance theory, the heat equation and the Atiyah-Singer theorem. C.R.C. press (1995). [HI Hida T.: Analysis of Brownian Functionals. Carleton. Math. Lect. Notes. 13 (1975). [H.K.P.S.] Hida T. Kuo H.H. Potthoff 3. Streit L.: White noise: an infmbe dimensional Calculus. Kluwer (1993). [LW] Ikeda N. Watanabe S.: Stochastic differential equations and diffusion processes. North Holland (1981). [J]Jaffe A,: Quantum harmonic analysis and geometric ionvariants. Adv. Maths. 143 (1999), 1-110. [J.L.O] Jaffe A. Lesniewski A. Osterwalder K.: Quantum K-theory. The Chern character. Com. Math. Phys. 118 (1988), 1-14. [J.L] Jones J.D.S. LQandreR.: LP Chen forms on loop spaces. In ”Stochastic analysis” Barlow M. Bmgham N. edit. Cambridge University Press (1991), 104-162. [Ku] Kusuoka S.: More recent theory of Malliavin Calculus. Sugaku 5. (1992), 155-173. [Ll] LBandre R.: Sur le theoreme de l’indice des familles. SBminaire de ProbabilitBs XXII. AzBma J, Meyer P.A. Yor M. edit. L.N.M. 1322 (1988), 348-413. [Lz]LBandre R.: Applications quantitatives et qualitatives du Calcul de Malliavin. In ”Col. Franco-Japonais” MBtivier M. Watanabe S. edit. L.N.M. 1322 (1988), 109-133. English translation: In ”Geometry of Random motion” Durrett R. Pinsky M. edit. Contemp. Maths. 73 (1988), 173-197. [L3]LBandre R.: A simple proof of a large deviation theorem. In ”stochastic analysis” Nualart D. SanzSolB M. edit. Prog. Prob. 32 Birkhauser (1993), 72-76. [L4] LBandre R.: Cohomologie de Bismut-Nualart-Pardoux et cohomologie de Hochschild entiere. SBminaire de Probabilitb XXX in honour of P.A. Meyer and J. Neveu. AzBma J Emery M. Yor M. edit. L.N.M. 1626 (1996), 68-100. [Ls]LBandre R.: Brownian cohomology of an homogeneous manifold. In ”New trends in Stochastic Analysis” Elworthy K.D. Kusuoka S. Shigekawa I. Edit. World Scientific (1997), 305-347. [LB]LBandre R.: Stochastic Adams theorem for a general compact manifold. Rev. Math. Phys. 13 (2001), 1095-1133. [L,] Mandre R.: Stochastic equivariant cohomology and cyclic cohomology. To appear Ann. Prob.

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[Ls]LBandre R.: Theory of distribution in the sense of Comes-Hida and Feynman path integral on a manifold. Inf. Dim. Ana., Quant. Probab. Rel. Top.6 (2003), 505-517 [Lg] LBandre R.: Hypoelliptic diffusion and cyclic cohomology. To appear in: ”Stochastic analysis”, R. Dalang, M. Dozzi, F. Russo edt. [Llo] Lkandre R.: White noise analysis, filtering equation and the Index theorem for families. To appear in: ”Infinite dimensional harmonic analysis”, H. Heyer, H. Sait8 edt. [Lo] Loday J.L.: Cyclic homology. Springer (1998). [MalIMalliavin P.: Stochastic Calculus of variations and hypoelliptic operators. In ”Stochastic analysis” K. It8 edi. Kinokuyina (1978), 155-263. [Maz] Malliavin P.: Stochastic Analysis. Springer (1997). [M.Q] Mathai V. Quillen D.: Superconnections, Thom classes and equivariant differential forms. Topology 25 (1986), 85-110. [Mo] Molchanov S.: Diffusion processes and Riemannian geometry. Russ. Math. Surveys. 30 (1975), 1-63. [N.R] Narasimhan M.S. Ramanan S.: The existence of universal connection. Amer. J. Math. 83 (1961), 535-581. [N.T] Nest R. Tsygan B.: Algebraic Index for families. Adv. Math. 113 (1995), 151-205. [Nil Nistor V.: Super-Connections and Non-Commutative Geometry. In ”Cyclic Cohomology and Noncommutative Geometry”. Cuntz J.R. Khalkhali Edt. Fields Inst. Comm. 17 (1997), 115-136. [Nu] Nualart D.: The Malliavin Calculus and Related Topics. Springer (1995). [0]Obata N. White noise analysis and Fock space. L.N.M. 1577 (1994). [Ou] Ouerdiane H.: Fonctionnelles analytiques avec conditions de croissance et application a l’analyse Gaussienne. Japan. J. Math 20 (1994), 187-198. [PI Perrot D.: A Bivariant Chern Character for families of spectral triples. Com. Math. Phys. 231 (2002), 45-95. [QI] Quillen D. Superconnections and the Chern character. Topology 24. (1985), 89-95. [Qz] Quillen D.: Algebra cochains and cyclic cohomology. Publi. Math. I.H.E.S. 68 (1988), 139-174. [St] Streit L.: An introduction to white noise analysis. In ”Stochastic analysis and applications in physics”. Cardoso A. SknBor R. Steit L. edit. Nato Asie Series. 449. (1994), 415-441. [Sz] Szabo R.: Equivariant cohomology and localization of path integrals in physics. Lec. Notes. Physics 63. Springer (2000). [Wall Watanabe S.: Analysis of Wiener functionals and its applications to heat kernels. Ann. Probab. 15 (1987), 1-39. [Waz]Watanabe S.: Stochastic Analysis and its applications. Sugaku 5 (1992), 51-71.

85

A Quantum Decomposition of L6vy Processes Yuh-Jia Lee* Department of Applied Mathematics National University of Kaohsiung Kaohsiung, TAIWAN 811

Hsin-Hung Shiht Department of Accounting Kun Shan University of Technology Tainan, TAIWAN 710

Abstract

The existence of the Lkvy white noise measure A as a Bore1 measure on X’ is proved. Then it is shown that a LQvy process can be represented by

Regarding X ( t ; x) as a linear operator in L(L,L’), the quantum decomposition of Lkvy process and the5enormalised” L6vy process are given, where L is the space of test Lkvy white noise functionals with dual space L‘.

AMS Mathematics Subject Classification 2000 : 60H40

1

Introduction

Let X = { X ( t ) : t E W) be a LBvy process, which is an additive process with stationary increaments on a probability space ( O , F , P )with X ( 0 ) = 0 almost surely. By the wellknown LBvy-Khintchine formula,

for T E Ri and --co < s uniquely expressed as

< t < +m, where f x

is called the Livy function of X which is

where p is a real constant, and ,B is a positive finite measure on W with u2 = p((0)). Conversely, for a given triple ( p ,a,p) with p, u E W and a positive finite measure p on W, there exists a Ldvy process X such that the equality (1.1) holds. We will say that X is a Lkvy process generated by the triple ( p ,u,p).

*Researchsupported by the National Science Council of Taiwan +Researchsupported by the National Science Council of Taiwan

86

In our previous paper, we have shown that the Lkvy white noise measures A exists and defined as a Borel measure on the space S' of tempered distributions under the the assumption that p has finite absolute first moment. Under this assumption, Gaussian, Poisson, Gamma processes and processes associated with Meixner class are included in our investigation, but the stable processes are excluded. We are desirable to extend the theory developed in [14] to stable processes. The study along this direction is by no means complete, in this paper we devote ourselves in improving our previous results in 114) without assuming that p has finite absolute first moment. The contents of the paper is arranged in the following manner. In section 2, the Lkvy process is represented as a function defined on K' and it is proved that the L6vy white noise measures always exists and is defined as Borel measure on the space K' of distributions without further assuming the moment conditions on p. It is always asked that if the measurable support is contained in S', we answer this question by given a criterion that the measurable support of a Lkvy white noise measure is contained in S'. In section 3, we introduced the space L of test Lkvy functionals, and the space L' of generalized Lkvy functionals, there the Lkvy white noise derivative and creation operator are defined and studied. In section 6, an connection between Lkvy white noise analysis and quantum probability theory are studied. There X ( t ; z) is regarded as a linear operator acting on L and the quantum decomposition of X ( t ; z) is given for the case where TI = 1.1 s_"udp(u) < 00; while if TI = 1.1 s_"udp(u) = ca, we obtain the quantum decomposition for the renomalized Lkvy process X ( t ) -.rlt. The former includes the cases such as Gaussian processes, Poisson processes, Gamma processes and the processes in the Meixner class; while the latter includes stable processes as special cases.

+

2

+

Levy white noise measure and a representation of LBvy processes

It is well-known that the sample paths of the Lkvy process X may be realized in the space D(P)of all right continuous functions #J having left limits and #J(O) = 0. Let 0 0be the set of all continuous functions 19 of R onto itself that are strictly increasing with 19(0) = 0. Then D(P)is a Polish space under the Skorohod metric p defined by

+

+

for any I$,$ E D(R), where kn(t) = 1 on -n 5 t 5 n, 0 at JtJ 2 n 1, -t n + 1 at n 5 t 5 n 1, and t n 1 at -n - 1 5 t 5 -n (see [lo]). Moreover, the Borel o-field B(!ID(W))of the metric space (D(W),p ) equals the c-field generated by all cylinder sets B of the form

+

B Now, let

+ +

= {#JE D(R) : (#J(tl),. . . , #J(tn))E E, E E f?(W)}, tl <

' ' '

< t,;

12 E

w.

i be a cylinder set measure in D(R) induced from the measure P by setting i(B)= P ( { w E R : ( X ( t l ; w ) , . . . ,X ( t , ; w ) ) E E } ) .

87

X

By using the Kol_mogorov extension theorem, has a unique extension to B(D(R)). Then (D(R),B(D(R)), A) is a probability space on which

Yx = { Y x ( t ;4) = 4(t) : t E w,4 E D(R)}

(2.2)

is a LBvy process with the same distribution as X on (O,F,P). Next, let K ( a ) be the space of all infinitely differentiable functions on R having compact supports in the interval [-a,a], a > 0. Then K ( a ) is a nuclear space [3]. Let K be the union of the spaces K ( a ) , endowed with the inductive limit topology and K’ the dual of K with the weak topology. Denote the K’-K pairing by (., .).

Proposition 2.1. T h e mapping T f r o m the Skorokhod space (D(R),p) into the space K’ by T(4) = i s injective and continuous, where i s the distributional derivative of 4.

4

4

Proof. First we point out that the mapping T is well defined since each element in D(R) is locally integrable on R. The injectivity of T is an immediate consequence of the Du Bois-Raymond lemma (see [4]).To prove the continuity of T, we assume that a sequence { A }of D(R) converges to 4 in D(R) under the Skorokhod metric p. Then there is a sequence {On} of 0 0 such that

{ :;

SUPtat IWt) - tl 0 SUP-r
+

0

for any r E M, as n -+ +oo (see [lo, ~2921).Observe that, for any 9 E K ( a ) , a > 0,

l(dn - 4,v)l I

J

-W

I M ~- +(t)~tli(t)~ ) dt t ~ t-)4 ( f i n ( t ) ) l l l i ( t ) ~ d+t J a

I 5

+W

-a

(

SUP

-a
14n(t)- 4(On(t))I) --a

1 4 ( ~ t-) ~) ~ t l i (dtt ) t

tli(t)Idt +

-a

14(%(t))- 4(t)llli(t)ldt.

So, by using (a) and (b) above, we deduce that (&, 9) converges to continuous.

(d, 9). Hence T is 0

Corollary 2.2. For any Borel set B of (’D(R),p), the image T(B) of B under the mapping above is also a Borel set of K’. In particular, T(D(R)) i s a Borel set of K’.

T as

Proof. We note that the space (!ID@), p) and K’ are both standard measurable spaces; that is, each of which is Borel isomorphic to a Borel subset of R viewed as a measurable space, because that (D(R), p ) is a Polish space and K’ is a projective limit of a sequence of dual countably Hilbertian spaces K(n)‘, n E M (see [7, 221). Then, by [7, Theorem 2.1.11 and 0 Proposition 2.1, the result immediately follows.

Let A be the probability measure on ( K ’ , B ( K ‘ ) )induced from that is, A ( E ) = X(T-’(E)), E E B(K’).

88

by the mapping T,

Consider the Fourier transform CA of the measure A, which is a complex-valued functional defined on K by CA(T)

=

Then we have the following result.

Theorem 2.3. For any

rj

kl

exp [i (x,1711 A ( d x ) .

in K , cA(17) = exP

[J_&,

fX(‘dt))dt],

where f x i s the LLvy function of X as in (1.i). Proof. From the definition of A,

Suppose that the compact support of

L

+00

4 ( t )+(t)d t =

-

+ is contained in the interval [-a,a]. Then

s_:

lim - - 12-00

rj(t) n-1

C

rj(tn,j)(4(tn,j+l) - 4 ( t n , j ) ) >

(2.4)

j=O

where tn,j = a(2j - n ) / n ,j = 0 , 1 , 2 , .. . , n - 1, for any n E the bracket of (2.3) by the formula (2.4), we see that n-1

cA(rj) =

ii% j=O

S,,

M. Replacing the integral in

exp[il)(tn,j)(Yx(tn,j+l; 4) - y X ( t n , j ; 4 ) ) 1i ( d 4 )

rn-1

17 The measure A will be called the LBvy white noise measure on K’ (generated by the triple (P, ( T I P ) ) . It is always asked under what conditions, the measurable support supp(A) c S’.An answer is given in the following theorem.

89

Theorem 2.4. [17] In order that the measurable support of the L6vy white noise measure A is contained in the space S‘ of tempered distributions on R, it is necessary and suficient that the complex-valuedfunction log@(.,q)(l),7 E K is continuous at zero in the topology of the Schwartz space S. It is easy to see that if the measure

p satisfies the moment condition

+m

1,

lulndp(u) < +co,

V n E M,

then the condition in the above theorem is satisfied so that A is supported by S‘. By MIX we denote the set of all random variables on the probability space (K’, B(K’), A) which is equipped with the metric p defined by

For any Y E M x , we denote its characteristic function by @ y . Let 8 be the class of all real-valued L’ n L2-functions 4 on R satisfying the following condition:

LI +m

fx(r+(s))l ds

< +co

for all r E R.

(2.5)

For each 4 E 8,it can be show that there exists a sequence (4%) of elements in the space K such that (i)

{a}converges to 4 in L1 n L2(R,dt);

(ii) The sequence {(,, &)} converges to a certain random variable Y in the space (MIX, p), such that

1, +W

@ y ( r )= exp(

f,y(r+(s))ds) for any r E R.

Denote Y by (., 4). In particular, when 6= l ( s , the t ~indicator , of (s, t], the characteristic function a(.,+)of (., 4) is exactly the same as the one in (1.1). Therefore the L6vy process X on (K’, B(lc’),A) can be represented in the following form:

3

Test and generalized LBvy white noise functionals

Segal-Bargmann transform Let Bb(R2) be the class of all bounded Bore1 subsets E of R: = R2\ {(t,O) : t E R} and define the product measure dv(t,u) = dpo(u) dt on B(R2). The well-known LBvy-It8 decomposition (see 1211) asserts that there exist a Poisson measure N ( E ;.) with intensity measure Y and a Brownian motion B(t) such that, for

90

b > a,

X ( b ) - X ( a ) = p ( b - a ) + o ( B ( b )- B ( a ) )

1 1

+i; % a < t j b n-'
U

[ u d N ( t , u )- 1+u2

where B = { B ( t ): t E R} is independent of the system of { N ( E ): E E ab(@)}. Let X be a positive measure on B(R2) defined by d X ( t ,u) = (1 u2)@(u) dt. Define a L2(K',A)valued function M on { E E l?(R2) : A(E) < +m} by

+

r+m

Then the system of { M ( E ; z) : E E f3(R2) with X(E) < $03; z E K'} forms an independent random measure with zero mean such that E[M(E)M ( F ) ]= X(E i l F ) for all E , F E B(R2). The multiple stochastic integral with respect to A4 was firstly introduced by It6 in [6]. In notation, we define In(g) by,

where g E sc((Rz)n,A@n),the closed subspace of Lz((R2)n,X@n) consisting of all symmetric complex-valued functions in L:((R2)", Awn) Then we have

We are ready to state Lkvy-It6 decomposition theorem for square integrable Lkvy white noise functional.

Theorem 3.1. [12]Let

'p

be given in L 2 ( K ' , A ) . Then h

( i ) there exist uniquely a series of kernel junctions & E L2((R2)"',A@"'), n E M u { 0 } , such that 'p is equal to the orthogonal direct sum CFy0@In(&). In notation, we write 'p (&).

-

(ii) L2(Ic',A) is isomorphic to the symmetric Fock space .Fs(Lz(R2, A)) over Lf(R2,A) bycarryingcp-(+,) i n t o ( f i q 5 O l f i q 5 1 ,..., fi& ,...).

For 'p E L 2 ( K ' , A ) ,say 'p (&), the Segal-Bargmann (or the 5'-) transform of complex-valued functional on L:(R2, A) by N

'p

is a

Then S-transform is a unitary operator maps from L2(K',A) onto the Bargmann-SegalDywer space .F1(L:(R2,A)) over L:(R2,X). Moreover, we have

91

where

D is the FrBchet derivative of Sv and 11 . llL(,,)(H)

denotes the Hilbert-Schmidt

(2)

operator norm of an n-linear functional on a Hilbert space H . The Segal-Bargmann transform enjoys an integral representation introduced in 1121. Let A = -d2/dt2 (1 t 2 ) be a densely defined self-adjoint operator on L2(R,d t ) and {en : n E No} be eigenfunctions of A with corresponding eigenvalues 2n 2, n E N U (0). {en : n E No} forms a complete orthonormal basis (CONS, in abbreviation) of L2(R2,dt). For m y p E W and q E L2(W,d t ) , define Iqlp := I A P q 1 p ( ~ , d and ~ ) let S, be the completion of the class (77 E L2(B,dt): )q), < +a}with respect to 1 . Jp-norm. Then S, is a real separable Hilbert space and we have the continuous inclusions:

+ +

+

S = I ~ p ~ o S p ~ S pL 2c (SWg, dct ) C S - , ~ S - p C S ' = ~ p > O S - pp ,> q > O .

+

Similarly consider the real Hilbert space L2(R,y), where dy(u) = (1 u2)dp(u)is a 0-finite measure on (R, a@)). Choose a CONS ((0, < I , . ..} of L2(R,y). Define a linear operator Ap densely defined on L2(R,y) by Ap Cn = (2n 2) Cn for n = 0,1,. . .. For each p 2 0, let Ep be the set of all C E L2(R,y) with l A ; C 1 + 2 ( ~ , < ~ ) +co. Then Ep is a real separable Hilbert space with the inner product (., .)p,p given by

+

d E .

and induced norm by I . lp,p := Denote by E-, the dual of Ep. Then E-, is isometrically isomorphic to the completion of L2(W,y) with the inner product (.,.)-,,p and I . I-,p-norm by

where we identify z E E-, with En(%, In) In in which (., .) is the E-p-€p pairing. Let E = lim Ep. Then E is a nuclear space with the dual E' = &-., E c L2(R,y) c +p>o E' forms a Gel'fand triple. Additionally, we assume that 1 E E'.

Remark 3.2.

J:z

(1) If the measure p satisfies the absolute moment condition of all order, i.e., lul" dp(u) < +co for all n E N, then we can apply the method of Gram-Schmidt orthogonalization to {1,u,u2,...} to obtain a CONS (<0,[1,. ..} of L2(W,y). In this case, 1 E E c E'.

(2) Let X be an a-stable process with 0 < (Y _< 1 such that d ~ ( u= ) ci

I u I ~ - ~l(-w,o)(u) d~ + ~2 2

~

l(o,+,)(u) ~ ~ du,

~ 1c2,

> 0.

We can choose a CONS {Q, ( 1 , . . .} of L2(R,y) by setting

where hn's are the Hermite functions on R. Then

and it is obvious that

E E E'

with ( ( u ) = un, u E R, for any n E N U (0).

92

For p E R,denote by Np the Hilbert space tensor product S, C3 EP with 1. Ip-norm defined by lei @ C j I p = lei\, I
N'

=

S' @ E'.

N c L2(Rz,A)c N' forms a Gel'fand triple. Moreover, the inclusions N c Np c N, c L2(R2,A) c N-, c N-, c N', p > q 2 0, are all continuous. Then

In the following, we denote for simplicity, where

I.I,:=I.I NZZ -

denotes the symmetric tensor product. In addition, we relabel the

CONS {en @

Cm : n,m = 0,1, ...} by {fo,fi ,...1,

in such a way that fo = e o @ and let Aj = I ( A @ A p ) f j bfor j E N U (0). We next proceed to construct the spaces of test and generalized functions as follows. For p E W and 'p E L2(S',A), define

and let L, be the completion of the class { 'p E L2(S',h): 1\'pl), < +co} with respect to 1) . Ilp-norm. Then L,, p E R,is a Hilbert space with the inner product ((., .)), induced by 11 . Il,-norm. For p , q E R with q 2 p , L, c L, and the embedding L, L, is of HilbertSchmidt type, whenever q - p > l/2. Let L = l&p,o L,. Then L is a nuclear space. L will serve as the space of test functions and the dual space L' of 13 the space of generalized functions. The members of L' are called generalized Lkvy white noise functionals. In this way, we obtain a Gel'fand triple L c L2(S',A) c L' and have the continuous inclusion:

-

L c L,

c L, c L2(S',h)c Lb c LL c 13'

=

9,, L;, ,p 2 q > 0.

In what follows, the dual pairing of L' and L will be denoted by ((.,.)) . Example 3.3. For 17 E Ic, 11(7 8 1) E L' since 17 @ 1 E integral with the kernel function h E L2(R2,A).

N', where 11(h) is the Lkvy-It6

For g E Lz(B2,A), it is easy to see that llEM(g)llp = e(1/2)lgg for any p > 0. Hence L, if and only if g E JfP,= for p > 0. We define the S-transform for F E 13' by

EM(g) E

S F ( g ) = ((F,& M ( 9 ) ) ) ,

9E N .

Annihilation and creation operators Let F E L, and ( E N-p,c,p E B. The Ggteaux derivative (d/dz)l,,o SF(.+zE) in the direction ( is an analytic function on N-p,c.In fact, by using the Cauchy integral formula and the characterization theorem [14],one can show that S - l ( (d/dz)l,=, S F ( . z l ) ) E Lp-z. Define a, F = S-'( (d/dz)l,_, S F ( . z < )).

+

+

Then we have

a, F

in

L,-1.

93

It is clear that 8, is continuous from C into itself. Its adjoint operator 8; is then defined from by ((8; F, p)) := ((F,8 ~ 9 ) ) for F E C‘ and p E

L.

8, is called the annihilation operator and 6’; is called the creation operator. It can be shown that, for p > 1/2,

Let Ap be a maximal Bore1 subset of W2 such that ( t , u ) E Ap. Then A(W2 \ A,) = 0 and A, C A, as p any t E W,whenever p ( { u } ) > 0. Let

u>

A=

p E N;p

Then A(W2

\ A) = 0.

Cj”=,Ifj(t,u)121fj1!p is finite for 5

q. We note that ( t , u ) E A, for

AP.

1/2

> 1/2, by

Define b(t,u)be the functional on Np,p

if ( t ,u ) E A; otherwise, b(t,u)= 0, where the sum in (3.6) is absolutely convergent in N,. It is easy to see that if F E L, ( p 2 2), then, for [A] almost all ( t , u ) E R2, 8(t,u)F =

&(,,+,

F

in L2(K’,A).

. . . , (tn,un)E W2 \ A, we have

and, for p > 1/2 and

( t l ,u l ) , -

A

(b(tl,ul)@.

..@ 6(t,,,u,,), 9) = g((tl,u1),. . . ,( t n , u n ) ) ,

g E NtF,

where (., .) is the N!&-Nfp pairing. If the L6vy white noise measure A is analytic, then L2(S’,A) includes square integrable be the space of the projective limit of {&;,,(SL)}for analytic functionals. Let &Ain(SL) L such that which &j,k(SL)consists of all analytic functionals p on S : z E S-,,=} < +m.

sup{l‘p(z)l e-(l’k)’+p

In this case supp(A) c S’, and, by [14, Theorem 2.71, &Ain(SL) c L2(S’,A). Then the creation and annihilation operators enjoy respectively the following integral representations.:

(ii) 8;

‘p

=

L?

uh(t,u ) p(. - ubt) d N ( t ,u ) -

94

(L?

1

u q t , u ) dv(t,u )

4

Quantum decomposition of LBvy processes

For a fixed p 2 0, denote by M , the class consisting of all functions h in N, so that the associated multiplication operator Mh, which is defined by M h ( g ) = h*g for g E Np,,,acts continuously frornJ& into Lt(R2,X),where h*(t,u)= u h ( t , u ) ,(t,u)E R2. For h EM,, let d r h be the differential second quantization of M h , i.e., for 91,. . . ,gn E N,,,, mh(g1G . . .

Ggn) =

G g2G . ‘6 gn + 91% ~ h ( g 2G) 93 G . . G gn + ...+ glG...Ggn-lGMh(gn) .

Mh(g1)

Now, let 8; be the linear operator on the linear space spanned by I n ( g l % . . . G g n ) , 91,. . . ,gn E N,,, and n E M, defined by 8; In(g1G .

For ‘P

N

( 4n )

E L,,4

’ 3g n ) =

In(drh(giG. . . G g n ) ) .

2 0, let

Let h 6 M,, p L. 0 and p E L, with q - p

(k

{%

2 1. Then

n=O ln(d’n,k))}w

k=O

is a Cauchy sequence in L2(K’,A). This leads us to the following

Definition 4.1. Define

It follows immediately from the Definition 4.1 that we have

where “sym” means “the symmetrization of ”. Moreover,

11%

(PI10

5

lllP11q1

where IlMhll is the operator norm of Mh. 8: is called the conservation operator indexed by h.

95

Theorem 4.2. Let h E M , with p 2 0 and 'p ( & ) E L, with q - p 2 1. Then 8, 'p, 8; 'p, and 8; 'p are in Lz(S',A). Moreover, for [A] almost all x E K', N

I l ( h ) ( x )Lp(X) = I n particular, when q 8 1 E M , for q

for [A] almost all x

ah ' f ( X )

+ 8; dz)+ 8; dx).

E K, and

E K', where 71 = p

+ $-',"

$-',"

(4.1)

u dp(u) < +03,

udp(u)

The identity (4.1) is called the quantum decomposition of the process I l ( h ) . If finite the identity (4.2) is called the quantum decomposition of the process (x,q).

71

is

Proof. (Sketch) We verify the identity (4.1) for 'p = Im(gBn) By the product formula [13], we have

+

I i ( h ) L ( g B n )= m( h, g ) L - l ( g o m - l ) L + l ( h G g B m )+mIm((hg)*%(g"-')) = ah Im(gBm) 8; Im(g@'") 8; Im(gBm).

+

+

The second assertion follows from the fact that

0

In the rest of this paper, we assume that there is a fixed positive number po such that for any q E K ,the associated operator by carrying g ( t , u ) into u q ( t ) g ( t , u ) , (t,") E R2, is continuous from Npo,c into Lz(R2,A). Then the definition of the conservation operator 8; can be extended to h E K 8 1 so that the related properties stated as above hold. For notational convenience, we identify 8; with The conservation operator can also be written as the following more familiar form: for q E K and 'p E L,

where the integral exists in the sense of Bochner (see[l4]). To derive the quantum decomposition for the L6vy process, we need the following lemma.

Lemma 4.3. Let 'p E C and the sequence {qn} of K converge to 4 in L1 n L 2 ( R , d t ) . Then

exists. W e denote such a limit by I i ( 4 ) 'p.

96

Proof. Obviously {8,,,+n,nlp} and {13,f,,,~[-,,,~ p} are convergent to 8, p and 8; respectively. On the other hand, since for q - po 2 1 and q E lC,

‘p

in L’

M

j=O m

which implies that

then we have

Apply the above estimation and use the quantum decomposition of I1(vn),the lemma follows immediately. 0 Finally apply the above lemma for q5 = l[o,tl, we derive the quantum decomposition of LBvy processes as follows.

97

Theorem 4.4. The "renormalized" L h y process X ( t ) -rlt is a continuous operator from L into L' and we have

( X ( t )- Tit) 'p =

& [ o , y 8 1 'p

+

+ ~ ~ [ o , c'pl @ l

I' 1:

'u.

q,,") a&(,,") P (1 + u 2 )dP(u)dt.

(4.3)

If r1 is finite (a case which excludes the a-stable process with 0 < a 5 l), we easily obtain the quantum decomposition for X ( t ) from the identity (4.3). Example 4.5. Let X = { X ( t ) : t E R} be an a-stable process with 0 < a 5 1 such that

+

(1 u2)d P ( U ) =

CI

IU('-~

l ( - m , o ) ( ~ du )

+ c2 u

~ ~ (-O ,~+ ~ ) ( dUu) ,

~ 1c2,

> 0.

R o m the Remark in the Section 3 we can obtain that for any g E N,, 3421)

= vm%7(t,4

l(O,+m)(.)

+ &-iT=g(t,U)

l(-w,O)(4

E

S@S.

Let p , q > 0 so that 1@u2 E S-,@S-,, and Ihlcl, 5 Const. Ihlq Ilcl, for any h , k E S,@S,. Then, for q E Ic,

s,

~ ~ q Ic(t,u)12dudt ( t ) ~

~ ~ q ](g (tt , u) ) I~2dX(t,u) = L

2

I171k I1@ 4

< Const.

-

- p . l?lP

11

@ u21-,

1~1,".

Thus, by Theorem 4.4,

References [l] Berezansky, Y. M., Kondratiev, Yu. G., Spectral Methods in Infinite Dimensional Analysis, (in Russian), Naukova Dumka, Kiev, 1988. English translation, Kluwer Academic Publishers, Dordrecht, 1995. [2] Doob, J. L., Stochastic Processes, Wiley, New York, 1953.

[3] Gel'fand, I. M., Vilenkin, N. Y., Generalized Functions, Vol 4. Academic Press, 1964. (41 Giaquinta, M., Hildebrandt, S., Calculus of Variations, Vol. I, Springer, Berlin, 1996. Potthoff, J., and Streit, L.: White Noise: An Infinite Dimen[5] Hida, T., Kuo, H.-H., sional Calculus, Kluwer Academic Publishers (1993). [6] ItB, K., Spectral Type of Shift Transformations of Differential Process with Stationary Increments, Trans. Amer. Math. SOC.81 (1956) 253-263.

98

[7] It6, K., Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces, CBMS-NSF Regionnnal Conf. series in Applied Math., Vol. 47, SIAM, Philadelphia, 1984 [8] Ito, Y., Generalized Poisson functionals, Probab. Theory Relat. Fields 77 (1988) 1-28. [9] Ito, Y., Kubo, I., Calculus on Gaussian and Poisson White Noises, Nagoya Math. J. 111 (1988) 41-84.

[lo] Jacod, J., Shiryaev, A. N., Limit Theorems for Stochastic Processes, Springer, Berlin, 1987. [ll]Kuo, H.-H., White Noise Distribution Theory, CRC Press, 1996.

[12] Lee, Y.-J., Shih, H.-H., The Segal-Bargmann Transform for Lkvy Functionals, J. h n c t . Anal. 168 (1999) 46-83. [13] Lee, Y.-J., Shih, H.-H., The Product Formula of Multiple Lkvy-It6 Integrals, Bull. Inst. Math. Academia Sinica 32, No. 2(2004), 71-95. [14] Lee, Y.-J., Shih, H.-H., Analysis of Generalized Lkvy White Noise Functionals, J. Funct. Anal. 211(2004), 1-70 [15] Lee, Y.-J., Shih, H.-H., A Characterization of Generalized L6vy Functionals, Quantum Information and Complexity, World Scientific, 2005, 321-339 1161 Lee, Y.-J., Shih, H.-H., The Adjoint of Lkvy White Noise Derivetive, in preparation, 2005 [17] Lee, Y.-J., Shih, H.-H., On the Support Property of Lkvy White Noise Measures, preprint, 2005 [18] Lytvynov, E. W., Polynimials of Meixner's Type in Infinite Dimensions-Jacobi Fields and Orthogonality Measures, J. Funct. Anal. 200 (2003) 118-149 [19] Meyer, P. A., Quantum Probability f o r Probabilistss, Lecture Notes in Math. 1538, Spring-Verlag, 1993. [20] Parthasarathy, K. R., A n Introduction to Quantum Stochastic Calculus, Basel/ Boston/ Berlin, Birkauser (1992).

[21] Sato, K., L h y Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999.

[22] Yamasaki, Y., Measures on Infinite Dimensional Spaces, Series in Pure Mathematics, Vol. V, World Scientific, Singapore, 1985.

99

GENERALIZED ENTANGLEMENT AND ITS CLASSIFICATION

T. MATSUOKA Faculty of Management of Administration and Information Tokyo University of Science, Suwa Chino City, Nagano 391-0292, Japan E-mail: [email protected] Recently the quantum entanglement has been focused t o study in quantum information theory. Several classifications of separable and entangled states have been studied extensively by many authors. Belavkin and Ohya (MO) gave a rigorous construction of quantum entangled state by means of Hilbert-Schmidt operator and gave a finer classification of separable and entangled state. In this article we review their entangling operator approach. The information degree of entanglement is applied for the entangled PPT state.

1. Introduction The entanglement was introduced by Schrodinger in 1935 out of the need to describe correlations of quantum states not captured by mere classical, statistical correlations which are always the convex combinations of noncorrelated states. There have been various studies of the entanglement in [1]-[8],in which the entangled state of two quantum system is defined by a state not written as a form

with any states pk and g k . A state written as above is called a separable state, so that an entangled state is a state not belonged to the set of all separable states in the naive definition above. However the mathematical characterization of entanglement is not yet fully understood except for some simple cases, for example pure states or low dimensional mixed states. Further it is obvious that there exist several types of correlated states, written as separable forms. Such correlated states have been considered in several contexts in the fields of quantum information and quantum probability such 100

as quantum measurement and filtering[9, lo], quantum compound states[ll, 121 and lifting[l3]. Belavkin and Ohya studied a rigorous operational structure of quantum entangled state by means of Hilbert-Schmidt operator (it is called the entangling operator) and gave a wider definition of quantum entanglement [14,15]. In this article we review how to construct the entangling operator H for a given compound state w according to [14, 151. We also discuss the information degree of entanglement given in [14, 151 and its application which is studied in [16]. The criterion is applied for entangled PPT states in the C 3 8 C 3 model of Horodeckis 171. We report some results obtained in [lS]. 2. Operational structure of quantum entanglement 2.1. Pure state Let K be a (separable) Hilbert space and B (K)be the set of all bounded linear operators on K. A normal state cp on B (K)can be expressed as

cp(B) = trxH*BH = trxBa, B

E B (K),

(1) where H is a linear Hilbert-Schmidt operator from an another Hilbert space 'H to K (i.e. IIH Ix1,)11~< +oo for any complete orthogonal system (CONS for short) (1x1,))in 'H) and a is the density operator corresponding to the normal state cp. This H is called the amplitude operator, which can always be considered on 7-t = K as the square root of the operator H H * (i.e. H = a+),and it is called just the amplitude if 'H is one dimensional space C, corresponding to the pure state cp ( B )= (
XI,

I<)

n

where {le,)} such that

is a CONS in 'H and IH,) is the orthogonal eigen-amplitudes IHn)E 'H, (H,IH,) = p,Sn,, with the eigenvalues p , of the 101

En

density operator u. Note that u = lHn)(Hnl is a Shatten decomposition, i.e., a one-dimensional orthogonal decomposition of u. If 'H is not one dimensional, then the dim% must not be less than rank H* = rank u and the dimension of the ran H * H coincide with that of the support ' H H ~ H . We can always equip 'H with the tilde operation "-" ('H-transposition) = J A * J given by an anti-unitary operator J on { / e n ) }in (2) as Jle,) = le,) and (JzI J y ) = (yI z) for z, y E 'H. Thus J is the standard complex conjugation in an eigen-representation of H * H (i.e. J ; C c,le,) H C c l e n ) , (em]xle,) =(en!A(emA(emIZ21(en)= (e,l A*Jem) with 71 = J A J ) such that H * H = H * H = H*H. Given the amplitude operator H from 'H to K ,one can define not only the state 'p but also the normal state

x

?1, ( A ) = t r x H X H * 5 t r x A p , A E B ('H)

(3) h l

on B ('H), where the corresponding density operator p is given by H * H , as the marginal of the following pure c m p o u n d state w

defined on B ('H @ K) on the composite system H ' @ K. The above bilinear form is uniquely expressed by the vector 10)E ' H @ K , which is given by (C @ 7]R) = (71HJC) for all I<) E 3.1 and 17) E K. In fact, it holds that

where { Ikn)} is a CONS in

K.

Lemma 2.1. For any A E B ('H) and B E B (K), w ( A@ B ) = t r X X H * B H = (RIA @ BlR)

holds. 102

Note that if H is represented by (2), then C le, 18 Hn) as follows

[R)becomes (R) =

n

n

n

n

n

Now let us move to the entangled mixed state and its entangling operator expression. 2.2. Classification of states via entangling operator

We can extend the argument of the normal state cp on B (K)to any normal compound state on B (3.1 @ K) . Let w be a normal compound state on

B(X@K)

is described by an amplitude operator V from an another Hilbert space 3 to 3-1 K ,satisfying the condition

VV* E B ('H @ K) , trwBKVV* = 1.

F can always be taken as 3 2 (3.1@ K ) e = Ee (3.1 @ K) for 6 = VV* with an anti-unitary operator J on eigen-representation of V*V. Then we can define an entangling operator H : 31 H F@ K uniquely by f i U = V , where

(C @ rll fit[) = ( J t @ I 1 ' H IJC) , VC E 3.1, rl E K ,c E F7

(6)

up to a unitary transformation U of 9. We have the following theorem.

Theorem 2.1. The normal compound state w in (5) can be achieved as an entanglement w ( A@ B ) = t r w x H * ( I @ B ) H = t r F B K H x H * ( I @ B ) h -

with its marginal states p = H * H on B (3.1) and 103

0

= t r 3 H H * on

(7)

B ( K ).

the eigen sepresentation of VV and the marginal P, and be a CONS

Any other choice of V is unitary equivalent to H in the minimal sapce Thus

Note that the lemma 2.1 can be obtained as a special case 3 = C in the theorem 2.1. Now let us see how the entangling operator can be used to classify the entangled states. We introduce the mapping 4* from B ('FI) to the predual B (K),by 4* ( A ) f trx ( A81)f3 = t r ~ H x H *for any A E B (X)and its dual mapping 4 from B (lc) to the predual B ('FI)* by

-

4 ( B )= trK:( I 8 B )f3 = H*( I 8 B ) H for any B

E B (K).Then the normal

compound state w is written as

-

w ( A8 B ) = trxA4 ( B )= trK:B4*( A ) .

The map B (E B (K))

(8)

4 ( B ):

4 ( B )= H* ( I 8 B ) H

-

(E B ('FI)*)

is the complete positive map (CP for short) written in the Steinspring form, $* * and the map A (E B ('FI))

4*

(x)

=

c(@I

(x1 .

8 0 HAH* (Ik) @ I > (E B (K)*)

k

-

is also CP written in the Kraus-Sudarshan form. Both maps 4 and $* are positive, but they are not necessarily CP, unless B(K)= B(K)or 104

B (3-1)

= B (H) (i.e., B (X) or

B ('FI) is Abelian). Both 4 and #* are called

complete co-positive. We define the true quantum entanglement [14, 151. Definition 2.1. The dual map q5* : B(7-l) + B ( K ) * of a complete copositive map 4 : B (K)--f B ('If)*, normalized as tr7-14 ( I ) = 1, is called the (generalized) entanglement of the state p z q5 ( I ) on B (3-1) to the state E $* ( I ) on B (K).The entanglement 4* is called true quantum if it is not complete positive. Let { I f n ) } and {le,)} be a CONS in 3 and H corresponding to the eigen-representation of V V * and the marginal p. In these eigenrepresentation the entangling operator H = V becomes H = C IHn) (en\=

-

c

k,n

Ifk

8 h k ( n ) )(en\, thus

v c Ivk) ( f k l

n

[en8 hk ( n ) )( f k l , where the

=

k

k,n

vectors Ihk ( n ) )E K: is defined by (en8 .I V I f k ) . Note that the eigen-basis of p is characterized by weak orthogonal condition tr?=@K IHm) (Hnl = And:

= (HmIHn)

(9)

where An are the eigenvalues of p. We summarize some notations for the sequel use and introduce a classification of quantum compound states. An entangled state w with its marginal states p and ISis expressed by a density operator 0 on H@X;that is, w (-) = tr .8, and 8 is written by the following forms due to the strength of the correlation between two marginal states in eigenrepresentation of p. (1) q-entang1ement:We denote true quantum entanglement by (i.e., 4: is not CP) and q-compound state by 0;

with weak orthogonal condition

The set of all true quantum entanglement by EQ. 105

4:

(2) d-entanglement: Let denote d-entanglement by #J$ and its compound state by e$. Then

n

n

with strong orthogonal condition

The set of all d-entanglement by

Ed.

(3) c-entanglement: The entanglement #J*is called c-entanglement if it has the same form as d-entanglement, but {on}are commutative. We denote c-entanglement by 4; and its compound state by 0;. €, is the set of all c-entanglements.

It is clear that &d and €, are belonged to the set of all not true quantum states. However there exist several important applications with quantum correlated state written as d-entanglement, such as quantum measurement and filtering, quantum compound state, and lifting. So that it is useful to classify the quantum separable state and the classical one. We show the necessary condition for separability. The state 0 is written as the convex combinations n

n

of tensor products of pure or mixed densities p n E B (X)* and un E B (K),, then n

n

which are given as the convex combinations of maps A H untrxApn and B H pntr&on. Such maps $* and are not only complete co-positive but also CP as it follows from the positive-definiteness of operator-matrics [A:Aj], VAi E B (X) , #J

So we have the following theorem. 106

Theorem 2.2. If a density operator I3 of a normal compound state w is separable, then both q$ and q$* are CP. The sufficient conditions in Theorem 2.2 will be discussed in a preparing paper [17]. 3. Degree of entanglement via quantum mutual entropy

In this section we review the classification of states by the information degree of entanglement in [14,151 and apply it to entangled PPT states in the C3 @ C3 model of Horodeckis[7].

3.1. Characterization of a pure state b y degree of entanglement Entanglement degree for mixed states has been studied by some entropic measures such as quantum relative entropy and quantum mutual entropy. As an example of such a degree was defined in [18]by the relative entropy S(I3,e0)rtr8(log13-log00) as

D (8) = min {S( 8 , B o ) ;80 E D},

(10)

where D is the set of all separable states. Since this measure has to take a minimum over all disentangled state, it is difficult to compute it analytically. Thus another degree of entanglement was introduced by Belavkin and Ohya[l4, 151.

Definition 3.1. Let w be the entangled state of p and a.Let q$ be the entanglement associated with w,and 86 is the density operator for w . (1) The quantum mutual entropy of p and a w.r.t $ is defined by Id ( p , 0) = trod (log Oc - logp @ 0). (2) The q-entropy of a is defined by 3 (a)= sup {I+ ( p , a) ;q6* (I)= a}. (3) D (q$;p,o) { S ( p ) S (a)} - I6 ( p , a) is called the degree of entanglement( DEM for short). 41 has stronger entanglement than 4 2 iff

=

+

(4) A compound state is said to be essentially entangled if D (4; p1a) < 0. 107

If the subalgebra A of B(IC) is abelian and u is a normal state on A, then s ( u ) is equal to von Neumann entropy = -traloga. Moreover if dimIC < +oo, then s ( u ) 5 logdimd, S ( a ) I IogdimK.

The above D (4;p, u) can be negative. If 8 on H @ IC is entangled pure state with the marginal states p, u,then von Neumann entropy S (8) = 0. Moreover, from the Araki-Lieb inequality:

1s ( P ) - s (0) I I s (8) Is (P) + s (4

(11)

we have S ( p ) = S (u).It follows

lo( p , G) = tr8 (log 8 - log p @ u) =s(p)

+

s(G)-

s(e)

= 2s (P>

That is, for entangled pure state, the q-entropy becomes twice of von Neumann (reduced) entropy [16].

Theorem 3.1. For a pure state w with the marginal state p and u, (1)

w i s separable ifl D (4;p , u) = 0 , and (2) w is inseparable (entangled) ifl D (4;p, u) = { S ( p ) + S (u)}< 0. A pure inseparable state can be called

-fr

as an essential entangled state.

3.2. Degree of entanglement for entangled PPT states The positive partial transpose criterion (PPT criterion for short) proposed by Peres [19] which is a necessary condition of separability but not a sufficient one generally. The IC-partial transpose operation of a compound density operator 8 on H @ K: is denoted by B T K such that

8-PK=(I@T)8, where dim 'FI @ K: is finite and T is the transpose operation on B (IC). For example, 8 is decomposable as 8 = C,,, [em)(e,l €3B,, where {le,)} is a standard base in H and B,, E B (K).Then

e

-

eTK

=

C le,) m,n

108

(en[@ B:,.

Definition 3.2. A compound density operator 8 on ?-I @ K is called a PPT state if BTK is positive. Non PPT state is simply is called NPT state. It is easy to show that the PPT condition is a necessary condition for a separable density operator. If a state 8 is separable and written as X k p k 8 crk, then eTK = x k X k p k 8 u; is positive because cr; is positive. However, in general, it is known that the converse statement does not hold. In the low-dimensional case (C2 @I C2 and C263 C3) Horodeckis showed that the PPT criterion gives a necessary and sufficient condition of separable states [ S ] . They also introduced the following state w, (.) = tr 8, as an example of the entangled PPT states on C3 63 C3[7]: a 5-a e, - 7- -2 IS+)(@+I ?e+ -e-, 2 1 a 1 5 7 where 1 la+)= - (1.1 63 el) le2 63 e2) le3 @I 4),

+

1

1

(1.)

8- = 3(

{ 11.)

(ell

63 1.2)

I 4 ( e 2 1 8 11.)

+

+

v5

8, = 2

+

1.2) (e21 8 (e3)(e3(

(e21

(ell

+ 1%)

( ~ 112.)8

+ le3) (e3(8 1.1)

( ~ 1 + )1.1

(ell 8 1 % )

(ell),

(e31),

,Ie2) , 1-23)) is a standard base in C3. The operator 0, has the following classification:

(1) 8, is a separable state for 2 5 a 5 3. (2) 8, is entangled but a PPT state for 3 (3) 0, is a NPT state for 4 < IY I 5.

< a 14.

We give another classification of 8, by means of the DEM [16]. First we remind the following theory [20].

Theorem 3.2. For a density operator p given as the convex combination n

n

of densities p n E B (‘H)*,the following inequality holds: n

n

The equality holds if pn Ipm f o r n # m.

109

The decomposition of 0, can be regarded as a convex combination of orthogonal states with the marginal states p and a given as p = (T = 1 5 (lei) ( e l l lez) (ezl le3) (%I). Then

+

+

1

D (6,; P, 0) = 5 { S (PI + s(a))- Ie, (P, a) 1

=

s(6,) - 5 { S (PI + s(4)

5-a 2 a a 5-a - -- log3 - - log - - -log 7

7

7

7

7

(13)

We obtain the change of the value D (Oa;p , a ) w.r.t. a as shown in Figure. I.

N P T '

Figure 1. D (Oa;p, 0)and classification of 0-

In Fig.1 we observe that the value a0 satisfying D ( O f f 0 ;p, a) = 0 is in between 3 and 4. Then we conclude an alternative classification of 8, as follows:

(1) 0, is not an essential entangled state for 2 _< a 5 ao. (2) 8, is an essential entangled state for a0 < a 5 5. The strength of entanglement can be read the change of the value

D (6,; p, a), that is the curve of the value. This degree can be used to find several models showing different types of entanglement which will be discussed in [17]. 110

References 1. R. F. Werner, Phys. Rev., A 40,4277 (1989). 2. C. H. Bennett, G. Brassard, C. Crepeau, R. Joza, A. Peres, W. K. Wootters Phys. Rev. Lett, 70,1895 (1993). H. Bennett, G. Brassard, S. Popescu, B. Schumacher, A. J. Smolin, W . Wootters Phys. Rev. Lett, 76,722 (1996). Ekert, Phys. Rev. Lett., 67, 661 (1990). Joza, B. Schumacher, J. Mod. Opt., 41, 2343 (1994). M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett., A 223, 1 (1996). M. Horodecki, P. Horodecki, R. Horodecki, ”Mized-State entanglement and quantum communication” in Quantum Information, Springer Tracts. in Modern Physics 173 (2001) B. Schumacher, Phys. Rev., A 51, 2614 (1993); Phys. Rev., A 51, 2738 (1993). V. P. Belavkin, Radio Eng. Electron Phys, 25,1445 (1980). V. P. Belavkin, Found. Phys., 24,685 (1994). M. Ohya, IEEE Info. Theory, 29,770 (1983). M. Ohya, Nuovo Cimento., 38,402 (1983). L. Accardi, M. Ohya, J . Appl. Math. Optim., 39,33 (1999). V. P. Belavkin, M. Ohya, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 4,No.2, 137 (2001). V. P. Belavkin, M. Ohya, Proc. R. SOC.Lond., A 458,209 (2002). T. Matsuoka, M. Ohya, t o appear in Proc. of Foundations of Probability and

3. C. K. 4. A. 5. R. 6. 7.

8. 9. 10. 11. 12. 13. 14. 15. 16.

Physics, AIP Proceedings. 17. A. Jamiolkowski, T. Matsuoka, M. Ohya, in preparation. 18. L. Henderson, V. Vendal, Phys. Rev. Lett., 84,2014 (2000). 19. A. Peres, Phys. Rev. Lett., 77,1413 (1996). 20. M. Ohya, V. D. Petz, Quantum Entropy and Its Use., Springer (1993).

111

A white noise approach to fractional Brownian motion David Nualart Universitat de Barcelona Facultat de Matembtiques Gran Via 585, 08007 Barcelona, Spain Abstract

We show that the derivative in time of a class of Volterra processes, that includes the fractional Brownian motion, is a Hida distribution. We review some facts about the divergence integral with respect to the fractional Brownian motion and we interpret this integral form the point of view of white noise analysis.

1

Introduction

The fractional Brownian motion (fl3m) is a centered Gaussian process B H = { B F , t 2 0 ) with the covariance function

RH ( t iS) = E(B,HB,")

=

1

-( 2

s

+ t2H ~ ~- It - ~

1

.

~

~

) (1)

The parameter H E ( 0 , l ) is called the Hurst parameter. The fractional Brownian motion has the following self-similar property: For any constant a > 0, the processes { a - N B z , t 2 0 } and { B y ,t 2 0} have the same distribution. From (1) we can deduce the following expression for the variance of the increment of the process in an interval Is, t]:

E (IB; - B f I 2 ) = It - ~

1

(2)

This implies that fBm has stationary increments. Furthermore, by Kolmogorov's continuity criterion, we deduce that fBm has a version with a-Holder continuous trajectories, for any a > H . For H = the process BY is an ordinary Brownian motion. However, for H # the process B H does not have independent increments, and, furthermore, it is not a semimartingale. Let r(n) := E [BB(B:+l - B:)]. Then, r(n) behaves as Cn2H-2,as n tends to infinity (long-memory process). In particular, if H > then IT-(.)\ = 00 (long-range dependence) and if H < then, Ir(n)l < 00 (short-range dependence).

i,

En

i,

i,

i,

En

112

The self-similarity and long memory properties make the fractional Brownian motion a suitable input noise in a variety of models. Recently, fBm has been applied in connection with financial time series, hydrology and telecommunications. In order to develop these applications there is a need for a stochastic calculus with respect to the B m . Because the fBm is neither a semimartingale nor a Markov process if H # new tools are required in order to develop a stochastic calculus. In this note we first construct the fBm in the white noise space and show that its derivative in time is a distribution in the sense of Hida. More generally, in Section 3 we introduce a class of Volterra processes that include fl3m and show that their time derivatives are Hida distributions. Section 5 reviews the approach to stochastic calculus for the fBm based on the Malliavin calculus, and in Section 6 we establish its relation with white noise analysis. We refer to Bender [4] and to Biagini et al. [5] for related works on this subject.

i,

2

White noise analysis and Hida distributions

In this section we present some preliminaries on white noise analysis. We refer to [9] and [13] for complete expositions of these notions. Let ( O , F , P ) be the white noise space. That is, R is the space of tempered distributions S’(R), 7 is the Bore1 0-field (with respect to the strong topology of S’(R)) and P is the Gaussian probability measure determined by E(exp(i(w,E)) = exP

1

(-5 Itell:)

9

for any rapidly decreasing function E E S(R), where 11.112 denotes the norm in L2(W). The pairing (w,Q can be extended using the norm of L 2 ( 0 ) to any function E 6 L2(R). In particular, the process Wt = ( . , l p t l ) is a Brownian motion. The trajectories of the Brownian motion Wt are nowhere differentiable almost surely. The theory of generalized functions introduced by Hida allows to give a rigorous meaning to the derivative To this aim we briefly recall the main aspects of this theory. Set (L2) = L2(R,F,P). Any random variable F E ( L 2 )can be developed into a series of multiple stochastic integrals:

%.

n=O

where fn E L2(Rn) is a symmetric square integrable kernel. Consider the operator A = x 2 1 and define its second quantization as

-& + +

n=O

113

Notice that the operators A and r ( A ) are densely defined in L2(R) and ( L 2 ) , respectively, and they are invertible and the inverse operators are bounded. For any p E R and F in the domain of I'(A)p,define

llFllp := ( E [ ( I ' ( A ) p F ) 2 ] ) 1 / 2 . If p 2 0, we denote by ( S ) , the space of random variables F E ( L 2 )such that llF/lp < 00 equipped with the norm Il.llp. If p < 0, we denote by ( S ) - p the completion of ( L 2 )with respect to the norm 1.1,. The projective limit of the spaces (S),, p 2 0, is called the space of test functions and is denoted by (S). The inductive limit of the spaces ( S ) - p ,p 2 0, is called the space of Hida distributions and is denoted by (S)*. Consider the Hermite functions defined for any n 2 0 as

E ~ ( =~ n-1/4(2nn!)-1/2e-f22~n(2), )

(3)

where Hn(s)is the nth Hermite polynomial, Hn(z)= (-l)ner2&e--22. The Hermite functions form an orthonormal basis of L2(B),and A& = (2n+2)En for any n 2 0 , because fk = f l C n - 1 - @&+I (with the convention (-1 = 0). Moreover, there exists a constant K > 0 such that

As a consequence of this estimate one can show that the derivative of the Brownian motion W'(t) belongs to (S)-,for any p > and W ( t )= (., &). The pairing (., bt) can be interpreted as the Wiener integral of the distribution bt, that is, I1(&).

A,

3

Derivative of a Volterra process

Ji

Consider a Volterra process of the form X t = K ( t ,s)dW,, where K ( t ,s) is a square integrable kernel such that K ( t ,s) = 0 if t < s, defined in the white noise probability space. That is, we assume that for any t 2 0

1'

K ( t ,s)2ds < 00.

In this section we will show that the derivative X ' ( t ) is an Hida distribution, under very general conditions on the kernel K ( t ,s). The covariance function of this process is

1

tAs

R(t,S) = E ( X , X , )

=

K ( t , r ) K ( s ,r ) d r .

We denote by &T the set of step functions on [0,TI. The space &m is the union of all ET. Let 'HTbe the Hilbert space defined as the closure of &T with respect to the scalar product ( I [ O , t ] 1[0,4),T , = R(t,s). 114

The mapping l ~ -~ + ,X ~ t can ] be extended to an isometry between ZT and the Gaussian space H T ( X ) associated with { X t , t E [O,T]}.We will denote this isometry by 'p --t X ( ' p ) . The random variable X ( ' p ) can be interpreted as the Wiener integral of 'p with respect to the Gaussian process X, that is, T X(p) = 'ptdXt, provided 'p can be identified as a function. Consider the mapping K - : & --+ L 2 ( 0 , T )defined by K-l[0,~1 = K (t , . ). This mapping can be extended to a linear isometry between 'FtT and L 2 ( 0 , T ) because

so

The operator K- allows to write the following transfer rule:

B(P) = W(K-'p),

(5)

for any 'p E Z T . We will make the following hypotheses on the kernel K ( t ,s): H1) K ( t ,s) is continuously differentiable on ( 0 < s < t < oo}, and its partial derivatives verify the following integrability condition:

for any 0 < E < T < 03. Moreover, t + continuous on (0, oo),for all 0 5 a < b. H2) The function k ( t ) =

Jot

StAbg(t,s ) ( t A b - s V a)+& is

K ( t ,s)ds is continuously differentiable on (0, 03).

Then, for any step function expression for the operator K -

'p

E &T, we can write deduce the following

In fact, it suffices to check (6) for an indicator function l[o,,], where a I T , and in this case, the right-hand side of (6) clearly vanishes for t > a , and for t < a yields

K ( T ,t ) -

lT%(.,

t)dr = K ( a ,t ) = (K-l[o,a])( t ) .

As a consequence, Hypothesis H1) implies that the set of continuously differentiable functions C1([O,T])is contained in ZT, and for any 'p E ZT, we have

115

We will denote by K+ the adjoint of the operator K - in L2(O,0o).That is, for any step function ‘p E ,&

Hence,

An alternative expression for the operator K+ is as follows

the right-hand side of (8) is In fact, if cp = l[0,~1, k’(t)l[o,al(t)+ For t > a this is we obtain

g ( t , r )[ l [ o , a ] ( r )- ~ [ o , a l ( t )dr. ]

s,” g(t,r ) d r = 6s,” K ( t , r )dr = ( K + l p a ]() t ) ,and for t < a t

K ( t ,S ) ~ S= (K+lp+1)(t). As a consequence, the space of continuously differentiable functions which are bounded together with its derivative Ci (0, co) is included in the domain of K+ and

IIcpllm + lIcp/lL

I(K+cp)(t)l I

J

0

l+-)l

dK

(t - r)dr.

(9)

The following result provides an estimate for K+&, where En are the Hermite functions defined in (3).

Lemma 1 For any n 2 0, K+&, is continuous o n (0,co) and

I(K+En)(t)I 5 ctn5/12, where Ct is bounded o n [&,TI, for all 0 < E

(10)

< T < 00.

Proof. From (8) we have

dK Clearly, k’(t)En(t) is continuous on (0,co)by hypothesis H2). On the other hand, we can write

116

and this is a continuous function of t by Hypothesis Hl), because it can be approximated uniformly on compacts of (0, m) by continuous functions. The estimates (9), (4)and Hypothesis H1) yield

aK I(K+tn) ( t ) ~5 ~k’(t)~ Iltnllm + I I G 0I I I z~ ( tJ ir)l( t - r ~

5 ctn5fI2, where Ct is bounded on [E, TI,for all 0 < E < T < co. rn The above lemma allows us to prove the following result.

Proposition 2 K(t, .) = K-lp,t] is differentiable from t E (0, co) into S’(R), and

Proof. Using the orthonormal basis of tions we obtain

L2(R)formed by the Hermite func-

Hence,

+

Using the estimate (10) we can bound each term of the above series by (2n 2)-2n5f6 Supt+
Proposition 3 Let F : (0,m) + S’(R) be a differentiable function. Then, 11( F ( t ) )= (., F ( t ) ) is a diflerentiable stochastic distribution process and [I1 (F(t))l’ = 11(F’@)).

Combining propositions 2 and 3 we see that X t is differentiable a t any t and X’(t) = Il(&0 K+), 117

> 0,

where St o K+ is the distribution defined by (& o K+,f) = (K+f) ( t ) , for any f E S’(R).In fact, m

M

d

&OK+ = C ( & o K + , E n ) E n = x (K+En)(t)E,=dtK-l[o,t] n=O

n=O

We can also write formally

X’(t) =

1-

(& o K+),W,’ds = (K+W’)(t).

(12)

Example Consider the Volterra process X associated with the kernel K ( t ,s ) = I’(H+i)-l(t-s)H-$, where H E (0,l). This process is related to the fractional Brownian motion with Hurst parameter H . Clearly, the kernel K ( t ,s) verifies properties H1) and H2). The operator K defined by (Kp) ( t ) = K ( t ,s)cp(s)ds coincides with the fractional integral operator I z ’ p . As a consequence, from (7) we obtain

Therefore,. the derivative of the Volterra process X is given by

that is, the derivative of this process is expressed as the fractional integral if

H > $ (or fractional derivative if H < k) of the white noise.

4

Fractional Brownian motion

The fE3m is a Volterra process (see, for instance, [8])

B? =

h’

K H( t ,s ) d w ~

associated with the kernel KH(t,s) given by

shere eg ius the normalizing constat c verifies

-(t, ~ K Hs) = CH(H- -1) ( t - S ) ~ - Z at 2 118

(I>

4-H,

and, as a consequence, Hypotheses H1) and H2) are satisfied. If H > the operator K- on 'HT can be expressed in terms of fractional

i,

integrals:

In this case, the scalar product of 'HT has the simpler expression

where CYH = H ( 2 H - l),and ZT contains the Banach space I H T ~of measurable functions 'p on [O,T]such that

We have the following continuous embeddings (see [14]):

LB(o,T))c I'HTI c ZT. For H < derivatives:

3,the operator K(K-'p) ( S ) = CHr(H

on 'HT can be expressed in terms of fractional

+1

2)S$-H

(D*IHUH-"p(U))

(16)

(S).

In this case, 'HT = I $ I H ( L 2 )(see 181) and 'HT contains functions which are y-Holder continuous, provided y > f - H . Using the fractional integration by parts formula (see [IS]) we obtain H - + u 1T - ~ ~ - - ' z'p(u))(s)

CHr(H - ; ) S H - + (Io+ C H r ( H ++H-++-H

'p(u)) ( s )

if if

H >f H
Hence, we obtain the following formula for the fractional white noise:

5

Stochastic calculus with respect to the fBm

Suppose now that u = { u t , t E [0,TI} is a random process. By the transfer rule ( 5 ) we can write

119

However, even if the process u is adapted to the filtration generated by the fBm (which coincides with the filtration generated by W ) , the process K-u is no longer adapted because from (14) and (16) we deduce that the operator K does not preserves the adaptability. Therefore, in order to define stochastic integrals of random processes with respect to the fBm we need anticipating integrals. In the case of an ordinary Brownian motion, the divergence operator coincides with an extension of It6's stochastic integral to anticipating processes introduced by Skorohod in [17] and Hitsuda [lo], [ll].Thus, we could use this anticipating integral in formula (17), and in that case, the integral JTutdBF coincides with the divergence operator in the Malliavin calculus with respect to the a m B H . The approach of Malliavin calculus to define stochastic integrals with respect to the fl3m has been introduced by Decreusefont and Ustunel in [8], and further developed by several authors (Carmona and Coutin [6], Albs, Mazet and Nualart [2], Albs and Nualart [3], Albs, Le6n and Nualart [l],and Hu [12]).

5.1

Stochastic calculus of variations with respect to fESm

We review here some basic facts of the Malliavin calculus for the fJ3m B H . Let S be the set of smooth and cylindrical random variables of the form

F =f(~~(41),...,B~(4n)),

(18)

where n 2 1, f E CF (Rn)(f and all its partial derivatives are bounded), and d'i E XT. The derivative operator D of a smooth and cylindrical random variable F of the form (18) is defined as the XT-valued random variable

The derivative operator D is then a closable operator from L2(fl) into L2(R;XT). We denote by D'v2 is the closure of S with respect to the norm

The divergence operator b is the adjoint of the derivative operator. That is, we say that a random variable in L2(f12;X~) belongs to the domain of the divergence operator, denoted by Dom 6, if

for any F E S. In this case 6(u)is defined by the duality relationship

120

ut6B2. for any F E D’i2. We will denote 6(u) by We have D1v2(’H~) CDom 6 and for any u E D ’ ~ 2 ( ’ H ~ )

E

(W2) = E (llull;T) + E ( ( D %

1

(20)

where (Du)* is the adjoint of ( D u ) in the Hilbert space ‘HT @ HT.

5.2

Divergence and symmetric integrals for H >

The following result (see [3]) provides a relationship between the divergence operator and the symmetric stochastic integral introduced by Russo and Vallois in [15].

Proposition 4 Let u = {ut,t E [0,TI} be a stochastic process in the space D’v2(3-I~).Suppose that

E (11~11;x,1+ llD~ll;xTlOIHT,)< and (21)

Then the symmetric integral tends to zero of (2El-l

J

so u t d B 2 , defined as the limit in probability as T

E

T H

-B ( ~ - ~ ) ~ ~ w ,

US(B~++T

0

exists and we have

Remark The symmetric integral can be replaced by the forward or backward integrals in the above proposition.

5.3

ItB’s formula for the divergence integral for H >

3,

If F is a function of class C2, and H > the path-wise Riemann-Stieltjes F ’ ( B f ) d B f exists for each t E [0,TI by the theory of Young [18]. integral Moreover the following change of variables formula holds:

F ( B F ) = F ( 0 )+

l

F’(B7)dBy.

Suppose that F is a function of class C2(R) such that

121

&.

where c and X are positive constants such that X < Then, the process F’(Bp) satisfies the conditions of Proposition 4. As a consequence, we obtain

=I’

F‘(Bf)SBf

+H

6’

F”(Bf)~~~-’ds.

(25)

Therefore, putting together (23) and (25) we deduce the following It8’s formula for the divergence process

A more general version of It8’s formula has been proved in [3]

5.4

Stochastic integration with respect to fBm for H <

The extension of the previous results to the case H

<

is not trivial and new

difficulties appear. For instance, the forward integral BpdBp in the sense of Russo and Vallois does not exists, and one is forced to use symmetric integrals. A counterpart of Proposition 4 in the case H < f and It6’s formula (26) has been proved in [l]for f - H if c < 2H - f . If H 5 one can show (see [7]) that

a.

4

a,

P ( B H E ‘FIT)

= 0,

and the space D1,2(‘FI~) is too small to contain processes of the form F’(BF). In [7] a new approach is introduced in order to extend the domain of the divergence operator to processes whose trajectories are not necessarily in the space ‘FIT. The basic tool for this extension of the divergence operator is the adjoint of the operator K- in L 2 ( 0 , T )that we have denoted by K+. Set 7f2 = (K-)-’ (K+)-’ (L2(0,T ) )and denote by Sx, the space of smooth and cylindrical random variables of the form

F = f(BH(41), where n 2 1, f E C p (Eln),and q5i E

. 1

BH(4n)),

(27)

‘FI2.

T

Definition 5 Let u = { u t ,t E [0,TI} be a measurable process such that Es, uTdt < 00. We say that u E Dom*b if there exists a random variable 6(u) E L 2 ( 0 ) such that f o r all F E Sx, we have

s,

E(u,K+K-D,F)dt = E ( 6 ( U ) F ) .

122

This extended domain of the divergence operator satisfies the following elementary properties:

1. Domb

c Dom*b, and 6 restricted

to Domb coincides with the divergence

operator.

2. If u E Dom*b then E ( u ) belongs to

XT.

3. If u is a deterministic process, then u E Dom*b if and only if u 6 Xi-. This extended domain of the divergence operator leads to the following version of It8’s formula for the divergence process, established by Cheridito and Nualart in [7].

Theorem 6 Suppose that F is a function of class C2(R) satisfying the growth condition (24). Then for all t E [O,T],the process {F’(B$’)lro,tl(s)} belongs to Dom*6 and we have

6

White noise analysis and divergence integrals

In this section we will introduce a general definition of the anticipating integral using the Wick product and the approach of white noise analysis (see [5] and [13]) and we will show that it includes the divergence integral studied in Section 5. A fundamental tool in white noise analysis is the S-transform. For any generalized random variable F E ( S ) * ,the S-transform is defined by

S F ( ( ) = ( F , e w ( ~ ) - ,~6~E S(R) ~ ~ ~ ~ ~ ) Here are some well-known which is well defined because e w ( ~ ) - * l ~ ~ E l (S). l~ properties of the S-transform. (i) The S-transform S F characterizes the generalized random variable F . (ii) For F,G E (S)* there exists a unique element F o G E (S)* such that for all 6 E S(R), S ( F o G ) ( t )= S F ( ( ) S G ( ( ) .This element is called the Wick product of F and G. (iii) Let a stochastic distribution process X : [0,TI Then S(Xl)(O = ( S X ( 0 ) : .

-+

(S)* be differentiable.

If X is a Volterra process satisfying conditions H1) and H2), then S X t ( 6 ) = (7). We say a stochastic distribution process X : [O,T]+ (S)* is integrable if for all 6 E S(R), S X ( 6 ) is measurable, S X ( [ ) E L1(O,T)and J:SXt(E)dt is the S-transform of a Hida distribution that will be denoted by SoTXtdt. A

(6,K ( t ,.))z and S X l ( 6 ) = (K+[)( t ) ,where K+ is given in

123

sufficient condition for the integrability of a stochastic distribution process is the fact that for all ( E S(R), S X ( ( ) is measurable and

ISXt(0I I W )exp

(;

IIAPFII;)

(294

for some p E M and L E L1(O,T). Suppose that u = { u t , t E [O,T]}and X = { X t , t E [O,T]}are stochastic distribution processes such that X is differentiable and the integral

1 T

ut O X ; &

exists. Then, we say that u is Wick integrable with respect to X . A particular case of this integral has already appeared in (12). The next proposition provides an example of a Wick integral with respect to a Volterra process.

Proposition 7 Suppose that X t = K ( t ,s)dW, is a Volterraprocesssuch that the kernel K ( t ,s) satisfies Hypotheses H1) and H2). Assume that the function Ct = Ik'(t)l

+

0

l$(t,r)I

(t - r)dr

belongs to L'(0,T). Let u = {ut,t E [O,T]}be a stochastic process such that EJ:u:dt < 00. Then u is Wick integrable with respect to X .

Proof. It suffices to show that Sut (K+E),satisfies (29a). But

1sutl =

. IE (utew(O-iIIcIIi)I 5 ( ~ u : ) feIIEIIz,

and on the other hand (K+E), can be bounded on any b i t e interval [O,T]by (9):

As a consequence,

Then,

sc (Eu:)' Ctdt I-4

< 00, and

n=O

g.

provided p > $, and (29a) holds for p > w On the other hand, the next proposition shows that the Wick integral (30) is an extension of the divergence integral we have introduced before using the techniques of Malliavin calculus.

124

Proposition 8 Suppose first that H > f , then any process u in the domain of the divergence is Skorohod integrable and

Also, if H < holds.

i, any process in the extended domain is Wick integrable and (31)

Proof. The fact that u is Wick integrable follows from Proposition 7. On the other hand, for any E E S(W), taking into account that OW(<)= (K-)-' [, and using the duality relationship (19) we obtain

which inplies the desired results because

References [l]Albs, E., Le6n, J. A. and Nualart, D. Stratonovich stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than 1/2. Taiwanese Journal of Mathematics 5 (2001) 609-632. [2] Albs, E., Mazet, 0. and Nualart, D. Stochastic calculus with respect to Gaussian processes. Annals of Probability 29 (2001) 766-801 [3] Albs, E. and Nualart, D. Stochastic integration with respect to the fractional Brownian motion. Stochastics and Stochastics Reports 75 (2003) 129152. [4] Bender, C. An It6 formula for generalized functionals of a fractional Brow-

nian motion with arbitrary Hurst parameter. Stochastic Process. Appl. 104 (2003) 81-106. [5] Biagini, F., (aksendal, B., Sulem, A. and Wallner, N. An introduction to white noise theory and Malliavin calculus for fractional Brownian motion. Preprint. [6] Carmona, P. and Coutin, L. Stochastic integration with respect to fractional Brownian motion. Ann. Institut Henri Poincare' 39 (2003) 27-68. 125

[7] Cheridito, P. and Nualart, D. Stochastic integral of divergence type with respect to the fractional Brownian motion with Hurst parameter H < Ann. Institut Henri Poincark To appear.

i.

[8] Decreusefond, L. and Ustiinel, A. S. Stochastic analysis of the fractional Brownian motion. Potential Analysis 10 (1998), 177-214. [9] Hida, T., Kuo, H. H., Potthoff, J. and Streit, L. White noise. A n infinitedimensional calculus. Mathematics and its Applications, 253. Kluwer Academic Publishers Group, Dordrecht, 1993. [lo] Hitsuda, M. Formula for Brownian partial derivatives. Second Japan-USSR Symposium on Probability Theory, Kyoto, 1972, 111-114. [ll]Hitsuda, M. Formula for Brownian partial derivatives. Proceedings of Faculty of Integrated Arts and Sciences. Hiroshima University 111-4 (1978) 1-15.

[12] Hu, Y. Integral transformations and anticipative calculus for fractional Brownian motions. Reprint. [13] Kuo, H. H. White noise distribution theory. Probability and Stochastics Series. CRC Press, Boca Raton, FL, 1996. [14] Pipiras, V. and Taqqu, M. S. Are classes of deterministic integrands for fractional Brownian motion on a interval complete? Bernoulli 7 (2001) 873-897. [15] Russo, F. and Vallois, P. Forward, backward and symmetric stochastic integration. Probab. Theory Rel. Fields 97 (1993) 403-421. [16] Samko S.G., Kilbas A.A. and Marichev 0.1. Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, 1993. [17] Skorohod, A. V. On a generalization of a stochastic integral. Theory Probab. Appl. 20 (1975) 219-233. [18] Young, L. C. An inequality of the Holder type connected with Stieltjes integration. Acta Math. 67 (1936) 251-282.

126

Adaptive Dynamics in Quantum Information and Chaos Masanori Ohya Department of Information Sciences, Tokyo University of Science, Noda City, Chiba 278-8510, Japan

Abstract

The concept of adaptive dynamics is explained and its application to quantum information and chaos is discussed. In particular, we explain how it is used to understand chaos.

1

Introduction

There exist several reports saying that one can observe chaos in nature, which are very much related to how one observes the phenomena in specified conditions, for instance, scale, direction. It has been difficult to find a satisfactory theory (mathematics) to explain such chaotic phenomena. An idea describing chaos of a phenomenon is to find some divergence of orbits produced by the dynamics explaining the phenomenon. However to explain such divergence from the differential equation of motion describing the dynamics is often difficult, so that one takes (makes) a difference equation from that differential equation, for which one has to take a certain time interval r between two steps of dynamics, that is, one needs a processing discretizing time for observing the chaos. In laboratory, any observation is done in finite size for both time and space, however one believes that natural phenomena do not depend on these sizes how small they are, so that most of mathematics (theory) has been made as free from the sizes taken in laboratory. Therefore mathematical terminologies such as "lim", "supll, "inf" are very often used to define some quantities measuring chaos, and many phenomena showing chaos have been remained unexplained. In the paper with Kossakowski and Togawa [8], we took the opposite position, that is, any observation will be unrelated or even contradicted to such limits. Observation of chaos is a result due to taking suitable scales of, for example, time, distance or domain, and it will not be possible in the limiting cases. In other words, it is very natural to consider that observation itself plays a similar role of "noesis" of Husserl and the mode of its existence is a "beingfor-itself", that is, observation itself can not exist as it is but it exists only

127

through the results (phenomena) of objects obtained by it. Phenomena can not be phenomena without observing them, so to explain the phenomena like chaos it is necessary to find a dynamics with observation. We claimed that most of chaos are scale-dependent phenomena, so the definition of a degree measuring chaos should depends on certain scales taken and more generally it is important to find mathematics containing the rules (dynamics) of both objects and observation, which will be called "Adaptive dynamics".

2

Adaptive Dynamics

In adaptive dynamics, it is essential t o consider in which states and by which ways we see objects. That is, one has to select phenomena and prepare mode for observation for understanding the whole of a system. Typical adaptive dynamics are the dynamics for state-adaptive and that for observable-adaptive. State-adaptive dynamics is that the dynamics of a system depends on a state at one instant in which the interaction is switched on, or that in a composite system the interaction depends on the instant state of at least one of subsystem. Examples of such adaptivity are seen in a compound state (or nonlinear lifting) studying quantum communication [6, 151 and in an algorithm solving NP complete problem in polynomial time with stochastic limit [l,2, 4,51. Observable-adaptive dynamics is that the dynamics of a system depends on observables to be observed and the measurement depends on an observable chosen. Examples of this adaptivity are used to understand chaos [8, 111 and examine violation of Bell's inequality [3]. There exists a deep relation between the adaptive dynamics and Information dynamics [7, 8, 161, which we will not discuss here. In this paper it is discussed how we can understand chaos in the adaptive scheme.

3

Entropic Chaos Degree

In quantum systems, for von Neumann entropy S ( p ) and quantum mutual entropy I ( p ; *) with a linear channel *, we define D ( p ; *) = S ( * p ) - I ( p ; *). Then it is easy to check that

where { E n }is the set of one dimensional projections giving a Schatten decomposition of p. Therefore the above quantity D ( p ; *) is interpreted as the complexity produced through the channel *. We apply this quantity D ( p ; *) t o study chaos even when the channel describing the dynamics is not linear. D ( p ; *) is called the entropic chaos degree in the sequel. In order to describe more general dynamics such as in continuous systems, we define the entropic chaos degree in C*-algebraic terminology. This setting

128

will not be used in the sequel application, but for mathematical completeness we will discuss the C*-algebraic setting. -Let (A,(S)be an input C* system and (Ale) be an output C* system; namely, A is a C* algebra with unit I and (S is the set of all states on A. We assume 2 = A for simplicity. For a weak* compact convex subset S (called the reference space) of 8,take a state cp from the set S and let cp =

J wdp, S

be an extremal orthogonal decomposition of cp in S, which describes the degree of mixture of cp in the reference space S.In more detail this formula reads

cp(4=

1

W(A)dP&), A E -4 s The measure p, is not uniquely determined unless S is the Choquet simplex, so that the set of all such measures is denoted by M , (S). Definition 1 The entropic chaos degree with respect to cp E S and a channel is defined by

Ds (p; *)

= inf

{

*

1

Ss (*w) d p ; p E M , (S)

where Ss (*cp) is the mixing entropy (or S-entropy) of a state cp in the reference space S . Before stating the theorem, we review the definition of the mixing S-entropy. For a state cp E S c G ( A ) ,put

D,(S)

=

where 6(p) is the delta measure concentrated on {cp}, and put

for a measure p E D,(S). Then the S-entropy of a state cp E S is defined as

ss(cp)=

{ ;2H

( P ) ; P E D,

(S)) when D, (S)# Q) otherwise

When S =8,Ds (cp; *) is simply written as D (cp; *) . This Ds (cp; *) contains the classical chaos degree and the quantum above. The classical entropic chaos degree is the case that A is abelian and cp is the probability distribution of a orbit generated by a dynamics (channel) *;

129

p=

Ekpkbk,

where

bk

is the delta measure such as

6k

(j)=

Then the classical entropic chaos degree is

1 (k=j) (k#.i)

0

Dc (9; *) = C p k S ( * b k ) k

with the entropy S . Summarize that Information Dynamics can be applied t o the study of chaos in the following way:

Definition 2 When p E S changes to *p, the entropic chaos degree associated to this state change(dynamics) * is given b y

{

Ds (p;*)= inf l S s ( * p ) d p ; pE Mv( S ) }. Definition 3 A dynamics

4

* produces chaos iff Ds (p;*) > 0.

Algorithm Computing Entropic Chaos Degree

In order to observe a chaos produced by a dynamics, one often looks at the behavior of orbits made by that dynamics, more generally, looks at the behavior of a certain observed value. Therefore in our scheme we directly compute the chaos degree once a dynamics is explicitly given as a state change of system. However even when the direct calculation does not show a chaos, a chaos will appear if one focuses to some aspect of the state change, e.g., a certain observed value which may be called orbit as usual. In the later case, algorithm computing the chaos degree for classical or quantum dynamics consists of the following two cases [Ill: N (1) Dynamics is given b y = Fb(z) with z E I = [a,b] c RN : First find a difference equation zn+l = F (zn)with a map f on I = [a,bIN C RN into itself, secondly let I = U k A k be a finite partition with A i f l A j = 8 (i # j ) . Then the state p(n)of the orbit determined by the difference equation is defined by the probability distribution ( p p ) ) , that is, p(n)= where for an initial value z E I and the characteristic function 1~

2

Now when the initial value z is distributed due to a measure v on I , the above

pin) is given as

( :J

The joint distribution p."n+')

) between the time n and n + 1 is defined by 130

or

Then the channel

*,

at n is determined by

(,,,+I)

: transition probability ==+ p(,+') = *,cp(,)

and the entropic chaos degree is given by

We can judge whether the dynamics causes a chaos or not by the value of Das

D > 0 u chaotic D =0 stable. This chaos degree was applied to several dynamical maps such as logistic map, Baker's transformation and Tinkerbel map, and it could explain their chaotic characters. This chaos degree has several merits compared with usual measures such as Lyapunov exponent as explained below. Therefore it is enough to find a partition {A,+}such that D is positive when the dynamics produces chaos. (2) Dynamics is given by pt = f:po on a Hilbert space: Similarly as making a difference equation for (quantum) state, the channel *, at n is first deduced from F:, which should satisfy p("+l)= *,p(,). By using this constructed channel, (a)we compute the chaos degree D directly according to the definition of ECD or ( p )we take a proper observable X and put x, = p(")(X),then go back to the algorithm (1). The entropic chaos degree for quantum systems has been applied to the analysis of quantum spin system and quantum Baker's transformation[9]. Note that the chaos degree D above does depend on a partition A taken, which is somehow different from usual degree of chaos. This is a key point of our understanding of chaos, which will be discussed in the following sections.

131

4.1

Logistic Map

Let us apply the entropy chaos degree (ECD) to logistic map. Chaotic behavior in classical system is often considered as exponential sensitivity to initial condition. The logistic map is defined by X,+l

= U2" (1- 2") , x , E

[O, 11,o I u54

The solution of this equation bifurcates as shown in Fig.1.

x,

a 3.2

3.4

3.6

3.8

4

Fig.1. The bifurcation diagram for logistic map In order to compare ECD with other measure describing chaos, we take Lyapunov exponent for this comparison and remind here its definition. Let f be a map on R, and let 20 E R. Then the Lyapunov exponent Xo(f)exponent of the orbit (3 = ( f " ( 2 0 ) ~ f o... o f ( x o ) : n = 0 , 1 , 2 , . . . ) is defined by

When f = (fi, f2,... , f m )is a map on R" and r0 E R". The Jacobi matrix J, = Df" ( T O ) at TO is defined by

132

0.5

.

0.4.

0.3

.

0.2

0.1 . 0:

3

3.2

3.6

3.4

3.8

4

Fig.2. Chaos degree for logistic map Then, the Lyapunov exponent A 0 (f)off for the orbit 0 = {f" (zo) ;n = 0 , 1 , 2 , .. .} is defined by ~0

(f)= log ,GI,

fik

= n-co lim ( p ~ )(~c= 1 , . . . ,m) .

Here, p: is the Ic-th largest square root of the m eigenvalues of the matrix J,J:. A 0 (f)> 0 + Orbit 0 is chaotic. A 0 (f)5 0 + Orbit 0 is stable.

The properties of the logistic map depend on the parameter a. If we take a particular constant a , for example, a = 3.71, then the Lyapunov exponent and the entropic chaos degree are positive, the trajectory is very sensitive to the initial value and one has the chaotic behavior. It is important to notice that if the initial value zo = 0, then z, = 0 for all n.

Tinkerbell map Let us compute the CD for the following two type Tinkerbell maps on I = [-1.2,0.4] x [-0.7,0.3].

133

fa

and

fb

A 0.5

a -0.5

-1

-1.5

3.2

3

3.6

3.4

3.8

4

Fig.3. Lyapunov exponent for logistic map

fa

(z'"')

= fa =

fb

(

(d"')= fb =

(zI.',zI"' 2

(Xi"')2

- (XI"')

+ a$) + CZXI"), 2 2 p X p + c 3 2 p + c 4 z p

),

(Xp',zI"')

((x?')'

where (z?',zI"')

-

+ c ~ x ? )+ C Z Z ~ ' ,2 2 P ) x p ' + bzp) + c42p) 1

E I , -0.4 I a I 0.9, 1.9 I b

5 2.9,

(CI, c2, c3, c4)

= (-0.3,

-0.6, 2.0, 0.5) and (z?),~?') = (0.1,O.l). Let us plot the points (z?',zI")) for 3000 different n ' s between 1001 and 4000. Fig.4 and Fig.5 are examples of the orbits of fa and fb in a chaotic domain.

134

7

$1 0.6

,

:::1

-1b

Fig.4. Orbits of

-0.7

fa.

1

Fig.5. Orbits of

135

fb

0.8 0.7

0.6 0.5 0.4 0.3 0.2 0.1

0 -02 -0.1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Fig.6. ECD of Tinkerbell map

1.4

a

fa

ECD r

1.2 1.o

0.8 0.6

0.4 0.2 0 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

Fig.7. ECD of Tinkerbell map

a

fb

The ECD of Tinkerbell map fa and f b are shown in Fig. 6 and Fig. 7. Here we took 740 different a’s between -1.2 and 0.9 and 740 different b’s between 1.9 and 2.9 with

--

100 100 (i = -120, -119,. . . -1, 0,1,. . . ’38’39) ( j = -70, -69,. .. ,-l,O, l , . .. ,28,29) n = 100000.

From the above examples it is seen that Lyapunov exponent and chaos degree have clear correspondence, moreover the ECD can resolve some inconvenient properties of the Lyapunov exponent as follows [lo]: 136

Lyapunov exponent takes negative and sometimes -m, but the ECD is always positive for any a 2 0. It is difficult to compute the Lyapunov exponent for some maps like Tinkerbell map f because it is difficult to compute f" for large n. On the other hand, the ECD of f is easily computed. Generally, the algorithm for the ECD is much easier than that for Lyapunov exponent.

4.2

ECD with memory

Here we generalize the above explained ECD to take the memory effect into account. Although the original ECD is based upon the choice of the base space C := { 1 , 2 , . . . ,N } , we here take another choice. Em, instead of C, is a new base space. On this base space, a probability distribution is naturally defined as

(n,n+l>, .,,n+m).The with its mathematical idealization, pioil...,,:= limn+cx,pioil nel A:, over Ern is defined by a transition probability,

..

~ j ~ i ~ , , . i , + ~ ~.i&,,~,j,j,,~

= P ( ~ I , ~* . z., i r n , i r n + l I . i ~ , . i ~ ,.. . ,.irn)~jo,jl...j,.

Thus it derives the ECD with m-steps memory effect,

It is easy to see that this quantity coincides with the original CD when m = 1. This memory effect shows an interesting property, that is, the longer the memory is, the closer the ECD is to the Lyapunov exponent for its positive part. The entropic chaos degree can be used to study some quantum chaos [9, 131.

5

Description of Chaos by Adaptive Dynamics

Our discussion of this section is based on a recent work (81, which was a trial to explain several chaos proposed in various experiments, so that our formulation can be applied t o dynamics in finite systems. First of all we examine carefully when we say that a certain dynamics produces a chaos. Let, us take the logistic map as an example. The original differential equation of the logistic map is

dx -_ dt

-x),O 5 u 5 4

U Z ( ~

137

with initial value xo in [0,1]. This equation can be easily solved analytically, whose solution (orbit) does not have any chaotic behavior. However once we make the equation above discrete such as x,+1 = ax,(l - X,),O

5 a 5 4.

This difference equation produces a chaos. Taking the discrete time is necessary not only to make a chaos but also to observe the orbits drawn by the dynamics. Similarly as quantum mechanics, it is not possible for human being to understand any object without observing it, for which it will not be possible to trace a orbit continuously in time. Now let us think about finite partition A={A,; k = 1,.. . , N} of a proper set I = [a,bIN c RN and equi-partition Be= {Bi;k = 1 , . . . ,N } of I . Here "equi" means that all elements Bg are identical. We denote the set of all partitions by P and the set of all equi-partitions by P e . In the section 4, we specified a special partition, in particular, an equi-partition for computer experiment calculating the ECD. Such a partition enables to observe the orbit of a given dynamics, and moreover it provides a criterion for observing chaos. There exist several reports saying that one can observe chaos in nature, which are very much related to how one observes the phenomena, for instance, scale, direction, aspect. It has been difficult to find a satisfactory theory (mathematics) to explain such chaotic phenomena. In the above difference equation we take some time interval r between n and n 1, if we take r -+ 0, then we have a complete different dynamics. If we take coarse graining to the orbit of xt for time interval r; 2, z $ xtdt, we again have a very different dynamics. Moreover it is important for mathematical consistency t o take the limits n + 00 or N (the number of equi-partitions)+ 00 , i.e., making the partition finer and finer, and consider the limits of some quantities as describing cham, so that mathematical terminologies such as "lim", "sup", "int" are very often used to define such quantities. Let us take the opposite position, that is, any observation will be unrelated or even contradicted to such limits. Observation of chaos is a result due to taking suitable scales of, for example, time, distance or domain, and it will not be possible in the limiting cases. It is claimed in [8]that most of chaos are scale-dependent phenomena, so the definition of a degree measuring chaos should depend on certain scales taken. Such a scale dependent dynamics is nothing but adaptive dynamics. Taking into consideration of this view we modify the definitions of the chaos degree given in the previous sections as below. Going back to a triple (A,(5, a! (G)) considered in Section 2 and we use this triple both for an input and an output systems. Let a dynamics be described by a mapping with a parameter t E G from (5 t o (5 and let an observation be described by a mapping 0 from (A,(5, a! (G)) to a triple (a,2,/3 (G)). The triple (B,T,p(G)) might be same as the original one or its subsystem and the observation map 0 may contains several different types of observations, that is, it can be decomposed as 0 = O,..Ol.Let us list some examples of observations.

sc-l)T

+

rt

138

For a given dynamics f = F ( c p t ) , equivalently, cpt = rtcp, one can take several observations. Example: Time Scaling (Discretizing): 0, : t -+ n, f ( t )--t cpn+l, so that d 3 = F ( c p t ) + cpn+l = F (pt) and cpt = r,*cp+ pn = l?Lcp. Here T is a unit time needed for the observation. Example: Size Scaling (Conditional Expectation, Partition): , of which have a cerLet ( B , % , p ( G ) )be a subsystem of ( A , G , a ( G ) ) both tain algebraic structure such as C*-algebra or von Neumann algebra. As an example, the subsystem ( B ,Z,@(G)) has abelian structure describing a macroscopic world which is a subsystem of a non-abelian (non-commutative) system (A,8,a ( G ) )describing a micro-world. A mapping 0 c preserving norm (when it is properly defined) from A to B is, in some cases, called a conditional expectation. A typical example of this conditional expectation is according t o a projection valued measure {Pk; PkPj = Pk&j = P;&j 2 0, CkPk = I } itssociated with quantum measurement (von Neumann measurement) such that 0~ ( p ) = CkPkpPk for any quantum state (density operator) p . When B is a von Neumann algebra generated by { P k } , it is an abelian algebra isometrically isomorphic t o Loo(0) with a certain Hausdorff space 0, so that in this case 0 c sends a general state cp to a probability measure (or distribution) p . Similar example of 0 c is one coming from a certain representation (selection) of a state such as a Schatten decomposition of p ; p = ( 3 ~ = p x k X k E k by one-dimensional orthogonal projections {&} associated to the eigenvalues of p with x k E k = I . Another important example of the size scaling is due to a finite partition of an underlining space 0, e.g., space of orbit, defined as 0 P (a)={Pk Pk n Pj = p k 6 k . j ( k ,j = 1,' ' ' N ), u= :, Pk = 0) .

5.1

Chaos degree with adaptivity

We go back to the discussion of the entropic chaos degree. Starting from a given dynamics cpt = rfcp,it becomes cpn = rLcp after handling the operation 0,. Then by taking proper combinations 0 of the size scaling operations like UC,0~ and Q p , the equation pn = r;cp changes to 0 (cpn) = 0 (I'Lcp), which will be written by 09, = Ol?~O-lOpor cpz = rLocpo.Then our entropic chaos degree is redefined as follows: Definition 4 The entropic chaos degree of I?* with a n initial state cp and observation 0 i s defined b y

where po is the measure operated b y 0 to a extremal decomposition measure of cp.

Definition 5 The entropic chaos degree of I?" with a n initial state cp i s defined by

139

D (9; I?*) = inf {Do(p;r*);O E SO} , where SO is a proper set of observations naturally determined by a given dynamics. In this definition , S O is determined by a given dynamics and some conditions attached to the dynamics, for instance, if we start from a difference equation with a special representation of an initial state, then SO excludes 0, and OR. Then one judges whether a given dynamics causes a chaos or not by the following way.

Definition 6 (I) A dynamics I?* is chaotic vation O iff Do (p;I?*) > 0.

for

a n initial state cp in a n obser-

(2) A dynamics I?* is totally chaotic for a n initial state p i f f D (cp; I?*) > 0. The idea introducing in this section t o understand chaos can be applied not only to the entropic chaos degree but also to some other degrees such as dynamical entropy, whose applications and the comparison of several degrees will be discussed in the forthcoming paper. In the case of logistic map, z,+1 = az,(l - z), = F (zn), we obtain this difference equation by taking the observation 0, and take an observation O p by equi-partition of the orbit space 0 = {z,) so as to define a state (probability distribution). Thus we can compute the entropic chaos degree as was discussed in Section 3. It is important t o notice here that the chaos degree does depend on the choice of observations. As an example, we consider a circle map =

fv(e,) = en + w

(mod 2 4 ,

where w = 2 m ( O < v < 1). If v is a rational number N / M , then the orbit (0,) is periodic with the period M . If v is irrational, then the orbit (0,) densely fills the unit circle for any initial value 00; namely, it is a quasiperiodic motion.

Theorem 7 Let I = [0,27r] be partioned into L disjoint components with equal length; I = B1n B2n . . . n BL. is rational number N / M , then the finite equi-partition , M ) implies Do (Q0; fv) = 0. (2) If v is irrational, then Do (00; fv) > 0 for any finite partition P={BI;). Note that our entropic chaos degree shows a chaos t o quasiperiodic circle dynamics by the observation due to a partition of the orbit, which is different from usual understanding of chaos. However usual belief that quasiperiodic circle dynamics will not cause a chaos is not at all obvious, but is realized in a special limiting case as shown in the following proposition.

(1)

P

If v

= { B k ;k = 1,. . .

Theorem 8 For the above circle map, if v is irrational, then D (00; f v ) = 0.

140

Such a limiting case will not take place in real observation of natural objects, so that we claim that chaos is a phenomenon depending on observations, surrounding or periphery, which results the definition of chaos as above. The details of this paper was discussed in [8] and will be discussed in [16].

References (11 Ohya M., Masuda N. “NP problem in quantum algorithm”, Open Systems and Information Dynamics, Vo1.7, No.1, 33-39, (2000)

[2] Accardi L., Sabbadini R. “On the Ohya-Masuda quantum SAT Algorithm”, in: Proceedings International Conference UMC’O1, Springer (2001)

[3] Accardi L., Imafuku K., Refoli M.:On the EPR-Chameleon experiment, Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 5, No. 1 (2002) 1-20 [4] Ohya M., Volovich I.V. “New quantum algorithm for studying NP-complete problems” , Rep. Math .Phys .,52, No. 1,25-33,(2003) and “Quantum computing and chaotic amplifier”, J.Opt.B, 2003. [5] Accardi L., Ohya M. “A Stochastic limit approach to the SAT problem”, Proceedings of VLSI 2003, and Open systems and Information Dynamics, 2004 [S] Accardi L., Ohya M., ”Compound channels, transition expectations, and liftings”, Appl. Math. Optim., Vo1.39, 33-59, 1999.

[7] Ohya M., Volovich I.V. “Mathematical Foundations of Quantum Information and Computation”, t o be published in Springer-Verlag [8] A.Kossakowski, M.Ohya and Y.Togawa (2003), How can we observe and describe chaos? Open System and Information Dynamics, 10(3):221-233 [9] K.Inoue, M.Ohya and 1.V.Volovich (2002) Semiclassical properties and chaos degree for the quantum baker’s map, J. Math. Phys., 43-2, 734-755. [lo] K.Inoue, M.Ohya and K.Sato (2000) Application of chaos degree to some dynamical systems, Chaos, Soliton & Fractals, 11, 1377-1385 Ill] M.Ohya: (1998), Complexities and their applications to characterization of chaos, International Journal of Theoretical Physics,Vol.37, No.1, 495-505.

[12] R.S.Ingarden, A. Kossakowski and M. Ohya: (1997), Information Dynamics and Open Systems, Kluwer Publ. Comp. [13] K.Inoue, M.Ohya and A.Kossakowski, A Description of Quantum Chaos, Tokyo Univ. of Science preprint (2002)

[14] Ohya M., Petz D.(1993), Quantum Entropy and its Use, Springer-Verlag 141

[15] Ohya M. (1983) On compound state and mutual information in quantum information theory, IEEE Trans. Information Theory, 29, No.5, 770-774. [16] Ohya M. Adaptive dynamics its use in understanding of chaos, TUS preprint

142

Micro-Macro Duality in Quantum Physics* Dedicated to Professor Tdeyuki Hida on the occasion of his 77th birthday Izumi Ojima

RIMS, Kyoto University, Kyoto, Japan

Abstract

Micro-Macro Duality means here the universal mutual relations between the microscopic quantum world and various macroscopic classical levels, which can be formulated mathematically as categorical adjunctions. It underlies a unified scheme for generalized sectors based upon selection criteria proposed by myself in 2003 to control different branches of physics from a unified viewpoint, which has played essential roles in extending the Doplicher-Haag-Robertssuperselection theory to various situations with spontaneously as well as explicitly broken symmetries. Along this line of thought, the state correlations between a system and a measuring apparatus necessary for measurements can canonically be formulated within the context of group duality; the obtained measurement scheme is not restricted to the quantum mechanical situations with finite number of particles but can safely be applied to quantum field theory with infinite degrees of freedom whose local subalgebras are given by type I11 von Neumann algebras.

1

Why & what is Micro-Macro Duality?

Vital roles played by Macro In spite of their ubiquitous (but implicit) relevance to quantum theory, the importance of macroscopic classical levels is forgotten in current trends of microscopic quantum physics (owing to the overwhelming belief in the ultimate unification at the Planck scale?). Without those levels, however, neither measurement processes nor theoretical descriptions of microscopic quantum world would be possible! For instance, a state w : B + C as one of the basic ingredients of quantum theory is nothing but a micromacro interface assigning macroscopically measurable expectation value

-

'Invited talk at International Conference on Stochastic Analysis, Classical and Quantum held in Nagoya in November 2004

143

w ( A ) to each microscopic quantum observable A E M. Also physical interpretations of quantum phenomena are impossible without vocabularies (e.g., spacetime 2, energy-momentum p , mass m, charge q, particle numbers n; entropy S, temperature T, etc., etc.), whose communicative powers rely on their close relationship with macroscopic classical levels of nature. Universality of Macro due to Micro-Macro duality Then one is interested in the question as to why and how macroscopic levels play such essential roles: the answer is found in the universality of “Macro” in the form of universal connections of a special Macro with generic Micro’s. To equip this notion with a precise mathematical formulation we -

F

introduce the notion of a categorical adjunction Q Z C which controls the E

mutual relations between [unknown generic objects Q (: microscopic quantum side) to be described, classified and interpreted] and [special familiar model C (: macroscopic classical side) for describing, classifying and interpreting], related by a pair of functors E(: c-iq) and F ( : q-c), mutually inverse up to homotopy I 3 E F , F E 5 I , via a natural isomorphism:

EaF(.) Q(w,E(a))

2 C(F(w),a) E(.)%

7

so that

an ‘equation’ E ( a ) w in Q to compare an unknown object w with controlled ones E ( a ) specified by known parameters a in C can be ‘solved’ to give a solution a F ( w ) which allows w to be interpreted in the vocabulary a in C in the context and up to the accuracy specified, respectively, by ( E ,F ) and (7,E ) . N

N

Abstract mathematical essence of “Micro-MacroDuality” can be seen in this notion of adjunction, whose concrete meanings are seen in the following discussion. What to be emphasized before going into details is the vast freedom in the choices of categories Q, C and functors E , F which are not to be fixed but adjusted and modified flexibly so that our descriptions are adapted to each focused context of given physical situations and to the aspects to be examined. This point should be contrasted to the rigidity inherent to the ultimate “Theory of Everything”. The simplest example of duality is given by the Gel’fand isomorphism,

CommC*Alg(M,C o ( M ) )N HausSp(M,Spec(%)),

(1)

between a commutative C*-algebra and a Hausdorff space defined by [cp*(x)](A) :=

[cp(A)](z) for [% 3 C o ( M ) ]2 [M 5 Spec(%) = {x : M + C ; x: character 144

s.t. x ( A B ) = x(A)x(B)}] and for A E %,a: E M . Through our discussion on the Micro-Macro duality below, we will encounter various kinds of fundamental adjunctions appearing in quantum physics as follows:

1) Basic duality between algebras/ groups and states / representations “Micro-Macro Duality” underlies “a unified scheme for generalized sectors based upon selection criteria” [14]proposed by myself in 2003 to control various branches of physics from a unified viewpoint. Extracted from a new general formulation of local thermal states in relativistic QFT (Buchholz, I 0 and Roos [2]), this scheme has played essential roles in my recent work to extend the Doplicher-Hag-Roberts superselection theory [5, 61 to recover a field algebra 5 and its (global) gauge group G from the G-invariant observable algebra % = SG and its selected family of states, according to which its range of applicability restricted to unbroken symmetries has been extended to not only spontaneously but also explicitly broken symmetries [15]. 2) Adjunction as a selection criterion to select states of physical relevance to a specific physical situation, which ensures at the same time the physical interpretations of selected states. This is just the core of the present approach to Micro-Macro Duality between microscopic quantum and macroscopic classical worlds formulated mathematically by categorical adjunctions: q-c

(generic:) Micro 2 Macro (: special model space with universality), c-q

where c -+ q (q -+ c ) means a c + q (q -+ c ) channel to transform classical states into quantum ones (vice versa). 3) Symmetry breaking patterns constituting such a hierarchy as unbroken / spontaneously broken / explicitly broken symmetries: the adjunction relevant here describes and controls the relation between [broken Z unbroken], playing essential roles in formulating the criterion for symmetry breakings in terms of order parameters. Through a Galois extension, an augmented algebra can be defined as a composite system consising of the object physical system and of its macroscopic environments including externalized breaking terms, where broken symmetries are “recovered” and the couplings with external fields responsible for symmetry breaking are naturally described.

4) If we succeed in extrapolating this line of thoughts to attain an adjunchomotopical dilation

?=! [stabilized hition between [irreversible historical process] erarchical domains with reversible dynamics] through enough controls over mutual connections among different physical theories describing different domains of nature, we would be able to envisage a perspective towards a

145

theoretical framework to describe the historical process of the cosmic evolution.

Basic scheme for Micro/Macro correspondence

2 2.1

Definition of sectors and order parameters

In the absence of an intrinsic length scale to separate quantum and classical domains, the distinctions between Micro and Macro and between quantum and classical are to some extent ‘independent’ of each other, admitting such interesting phenomena as “macroscopic quantum effects”. Since this kind of ’‘mixture”’ can be taken as ‘exceptional’, however, we put in parallel micro//quantum//non-commutative and macro//classical//commutative, respectively, in generic situations. The essence of Micro/Macro correspondence is then seen in the fundamental duality between non-commutative algebras of quantum observables and their states, where the latter transmit the microscopic data encoded in the former at invisible quantum levels into the visible macroscopic form. While the relevance of duality is evident from such prevailing opposite directions as between maps cp : 8 1 + U2 of algebras and their dual maps of states, cp* : Ea2 3 w +-+ cp*(w) = w o cp E Enl, their relation cannot, however, be expressed in such a simple clear-cut form as the Gel’fand isomorphism Eq.(l) valid for commutative algebras, because of the difficulty in recovering algebras on the micro side from the macro data of states. The essence of the following discussion consists, in a sense, in the efforts of circumventing this obstacle for recovering Micro from Macro. Starting from a given C*-algebra U of observables describing a Micro quantum system, we find, as a useful mediator between algebras and states, the category Rep, of representations 7r = ( 7 r , f i r ) of U with intertwiners T , T7r1(A)= 7r2(A)T (VA E a), as arrows E Rep,(nl,n2), which is nicely connected with the state space E, of U via the GNS construction: w E 1:l up to (7rw,4j,) E Rep, with 0, E 4 , s.t. w ( A ) = (0, I r,(A)R,)

E, unitaFequiv. ( V A E U) and T,(U)R, = jj,. Two representations ~ 1 , 7 7 3without (nonzero) connecting arrows are said to be disjoint and denoted by 7r1 A 7r2, i.e., def 7r1 b 7r2 e Repm(n1,nz) = (0). The opposite situation to disjointness can be found in the definition of quasi-equivalence, 7r1 = 7r2, which can be simplified into (: unitary equivalence up to multiplicity) u 7 r 1 ( 8 ) ” N “2(U)” u C ( T 1 ) = c ( r 2 ) u W*(7r1)*= W*(7r2)*.

7r1 M 7r2

To explain the central support c(7r) of a representation 7r, we introduce the universal enveloping W*-algebra U** N 7ru(U)” := W*(U) of C*-algebra U which contains all (cyclic) representations of U as W*-subalgebras W*(7r):= 146

T(%)” c W*(%). In the universal Hilbert space fj, := $ w E ~ m W*(U) f j w , and W*( T ) are realized, respectively, by the universal representation (T,, fj,), T , := & , E ~ m ~ w , and by its subrepresentations T ( A ) := P(T)T,(A) rp(,) (VA E a) in fj, = P ( T )with ~ ~P ( T )E W*(%)’.W*(%) is characterized by universality via adjunction,

W*Alg(W*(%), M ) N C*Alg(%,E ( M ) ) , between categories C*Alg, W*Alg of C*- and W*-algebras (with forgetful functor E to treat M as C*-algebra E ( M ) forgetting its W*-structure due to the predual M , ) with a canonical embedding map % E(W*(!Z)), so that any C*-homomorphism V p : % + E ( M ) is factored p = E ( $ ) o qa through qa with a uniquely existing W*-homomorphism $I : W*(U) -+ M :

z

U 7%

1 0

E(W*(%))

\vv

i;;)

E(M)

In this situation, the central support C(T)of the representation T is defined by the minimal central projection majorizing P ( T )in the centre 3(W*(%)) := W*(%) n W*(%)’of

w*(%).

i) Basic scheme for Micro-Macro correspondence in terms of sectors and order parameters: The Gel’fand spectrum Spec(S(W*(a)))of := W * ( %n ) I+’*(%)’ can be identified with a factor the centre 3(W*(%)) h

spectrum % of 8: h

Spec(S(W*(M)))N % := Fa/

=: factor spectrum,

defined by all quasi-equivalence classes of factor states w E Fa (with trivial centres ~ ( W * ( T:= ~W ) )* ( r wn) W * ( T ~= ) ’Cln, in the GNS representations ( ~ , , f j ~ ) ) . Definition 1 A sector of observable algebra U is defined by a quasiequivalence class of factor states of %.

In view of the commutativity of 3(W*(%)) and of the role of its spectrum, we can regard

-

0

0

Spec(S(W*(U)))N % as the classifying space of sectors to distinguish among different sectors, and

3(W*(%)) as the algebra of macroscopic order parameters to specify sectors.

147

Then the map h

Micro:

8*2 Em -H Prob(G) c LOO(%)* : Macro, h

defined as the dual of embedding 3 ( W * ( % ) ) N LOO(%)L) W * ( a ) can , be interpreted as a universal q-+c channel, transforming microscopic quantum states E Ea to macroscopic classical states E P r o b ( B ) identified with probabilities. This basic q + c channel, h

Em 3w

-

P~ = w” t3(wym))EE3(w*(m))= M 1 ( s p 4 3 ( W * ( n ) ) ) = ) Prob(G),

describes the probability distribution pu of sectors contained in the central decomposition of a state w of M: h

82A

-

w ” ( x a ) = pu(A)= Prob(sector E

A I w),

where wll denotes the normal extension of w E Em to W * ( 8 ) .While it tells us as to which sectors appear in w , it cannot specify as to precisely which representative factor state appears within each sector component of w . ii) [MASA]To detect this intrasectorial data, we need to choose a m a imal abelian subalgebm (MASA) n of a factor iM,defined by the condition = n 2 L m ( S p e c ( n ) ) . Using a tensor product iM@% (acting on the Hilbert-space tensor product c(7r)fi, @ L2(Spec(%)))with a centre given by

3(iM @ n)= 3(m)

n= 1 @ L m ( S p e c ( n ) ) ,

we find a conditional sector structure described by spectrum S p e c ( n ) of a chosen MASA n. iii) [Measurementscheme as group duality] Since the W*-algebra n is generated by its unitary elements U(n), the composite algebra M@% can be seen in the context of a certain group action which can be related with a coupling of iM with the probe system 8 as seen in my simplified version [14] of Ozawa’s measurement scheme [17]. To be more explicit, a reformulation in terms of a multiplicative unitary [l]can exhibit the universal essence of the problem. In the context of a Hopf-von Neumann algebra M ( c B ( f i ) )[7]with a coproduct r : M + M @ M ,a multiplicative unitary V E U ( ( M @ M * ) - )c U ( f i @ fi) implementing I?, r(o) = V * ( l 8 z ) V , is characterized by the pentagonal relation, v 1 2 & 3 h 3 = h 3 & 2 , on fi @ fi 8 3,expressing the coassociativity of r, where subscripts i ,j of Vij indicate the places in fi@fi@ fi on which the operator V acts. It plays fundamental roles as an intertwiner, V(X @ L ) = (A @ X)V, showing the quasi-equivalence between tensor powers X(w) := (i @ w ) ( V ) E & a of the regular representation X : M , 3 w generalized Fourier transform, X(wl * w2) = X(wl)X(w2), of the convolution

-

148

algebra M,, w1 * w2 := w1 8 w2 o I?. On these bases the duality for Kac algebras as a generalization of group duality can be formulated. In the case of M = Lw(G,dg) with a locally compact group G with the Haar measure d g , the multiplicative unitary V is explicitly specified on L2(G x G) by (Vt)(s, t ) := t ( s , s - l t )

for

4 E L2(G x G ) ,s,t E G ,

(2)

or symbolically in the Dirac-type notation, V(S,t ) = Is,s t ) .

(3)

-

Identifying M with the Hopf-von Neumann algebra Lw(G) = % corresponding to G := U(%)given by the character group of our abelian group U(%) (assumed to be locally compact), we adapt this machinery to the present context of the MASA ‘Jz, by considering a crossed product M G := [C 8 A(G)”] v a(M) [9] defined as the von Neumann algebra generated by C 8 A(G)” = C 8 6 and by the image a(M) of M under an isomorphism a of M into M €3 Lw(G) N Lm(G,9X) N M 8 ‘Jz given by [a(B)](y) := Ad,(B) = $,B4;, y E G, B E M where 4, is an action of y E G on M. By definition, 332 ) a L G = 9X8 6 is evident for the trivial G-action L with L(M)= M. The crossed product M )aor G is generated by the representation 4(V) = l G d E ( y )8 A, of V on L 2 ( M )8 L2(G) with the spectral measure E ( A ) = E ( x ~of) n (for Bore1 sets A in Spec(%)) defined by the embedding homomorphism E : % LC”(G) M of % into M, as seen from ( w €3 i)(4(V)) = A(E*w) E C 8 6 and (i 8 n)(+(V)) = lGdE(y)R(Xy) E a(M). The action of 4(V) corresponding to Eq.(3) can be expressed by

-

4(V)(t, 8 Ix))= t, €3

Irx)

for Y , X E G ,

(4)

satisfying the modified version of the pentagonal relation, 4(V)l24(V)13V23 = &34(v)12, or equivalently, V234(V)12V& = 4(V)124(V)13. Under the assumption that U(’Jz) is locally compact, the spectral measure E constitutes an impn’mitivity system, &(E(A))d; = E(yA), w.r.t. a representation 4 of G on L 2 ( M ) ,from which the following intertwining relation follows: 4(V)(4, 8 I) = (4, 8 A,)+(V), for y E G. While the role of a multiplicative unitary is to put an arbitrary representation p in quasiequivalence relation M with the regular representation A by tensoring with A: ,o8A U,(L8 A)UJ M A, the above relation allows us to proceed further to 4 M 4 ( V ) ( $ 8 L)4(V)*= 4 8 A U,(L 8 A)u; M A.

=

The important operational meaning of Eq.(4) can clearly be seen in the case where G is a discrete group which is equivalent to the compactness of the group U(’Jz)in its norm topology (or, the almost periodicity of functions on

149

it). In the present context of group duality with G as an abelian group generated by Spec(!Yt),the unit element L E G naturally enters to describe the neutral position of measuring pointer in addition to Spec(’JI),in contrast to the usual approach to measurements. Then Eq.(4) is seen just to create the required correlation (“perfect correlation” due to Ozawa [18]) between the states [, of microscopic system fm to be observed and that 17) of the measuring probe system (n coupled to the former: +(V)(C,@ J L ) ) = I, 8 17) for V y E G. Applying it to a generic statel E = CrEG cy&, of fm,an initial uncorrelated state E @ IL) is transformed by 4(V)to a correlated one:

The created perfect correlation establishes a one-to-one correspondence between the state [, of the system fm and the measured data y on the pointer, which would not hold without the maximality of !Yt as an abelian subalgebra of fm. On these bases, we can define the notion of an instrument 3 unifying all the ingredients relevant to a measurement as follows:

In the situation with a state W E = ( (1 (-)Q of fm as an initial state of the system, the instrument describes simultaneously the probability p(A1wc) = 3 ( A l q ) ( 1 )for measured values of observables in !Yt to be found in a Bore1 set A and the final state J(Alwe)/p(Alwc) realized through the detection of measured values [17]. While this measurement scheme of Ozawa’s is formulated originally in quantum-mechanical contexts with finite degrees of freedom where fm is restricted to type I, its applicability to general situations without such restrictions is now clear from the above formulation which applies equally to non-type I algebras describing such general quantum systems with infinite degrees of freedom as QFT. Since instruments do not exclude “generalized observables” described by “positive operatorvalued measures (POM)”, it may be interesting to examine the possibility to replace the spectral measure d E ( y ) with such a POM as corresponding to a non-homomorphic completely positive map for embedding a commutative subalgebra into fm. In what follows, the above new formulation will be seen to provide a prototype of more general situations found in various contexts involving sectors, such as Galois-Fourier duality in the DHR sector theory and its extension to broken symmetries with augmented algebras (see below). It is important there to control such couplings between Micro (fm) and Macro (‘TI as ‘Note that any normal state of !M in the standard f o r m can be expressed as a vectorial state without loss of generality.

150

measuring apparatus) as 4(V)E M M GI whose Lie generators in infinitesimal version consist of Ai E M and their “conjugate” variables to transform G 3 x F+ yix E G. This remarkable feature exhibited already in von Neumann’s measurement model, is related with a Heisenberg group as a central extension of an abelian group with its dual and is found universally in such a form as Onsager’s dissipation functions, (currents) x (external forces), as a linearized version of general entropy production [ll],etc. To be precise, what is described here is the state-changing processes caused by this type of interaction terms 4(V)between the observed system M and the probing external system ‘32, with the intrinsic (= %zperturbed”) dynamics of the form e r being neglected. While the validity of this approximation is widely taken for granted (especially in the context of measurement theory), the problem as to how to justify it seems to be a conceptually interesting and important issue which will be discussed elsewhere. iv) [Central measure as a c+q channel] Here we note that, from the spectral measure in iii), a central measure p is defined and achieves a central decomposition of M 8 % = L”(Spec(%), M) = J&,c%) MM,dp(a), where p ( A ) := wo(E(A))with wo a state of % supported by Spec(%) being faithful to ensure the equivalence p ( A ) = 0 e E ( A ) = 0. A central measure p is characterized as a special case of orthogonal measures by the following relations according to a general theorem due to Tomita (see [3] Theorem 4.1.25): for a state w E Ea of a unital C*-algebra U there is a 1-1correspondence between the following three items, 1) (sub)central measures p on En s.t. w = Ell u’dp(w’) and [JEs,s w’dp(w’)] [Jsu’dp(w’)] for V A : Bore1 set in En, 2) W*-subalgebras B of the centre: B c 3(W*(7rW)) = n 7r,(U)”, 3) projections P on 4, s.t. PR, = R,, P7r,(U)P c {PT,(Q)P}’. If p , B, P are in correspondence, they are related mutually as follows:

1. 93 = {P}’ n 3(W*(r,)); 2. P 3.

=

[BR,];

,(a,&. . .a,)

= (Owl7rw(A1)P7rU(A2)P.. . P~F,(A,)R,), where C ( E a )for A E U denotes a map A ( q ) = cp(A)for ‘p E Ea;

-

aE

4. B is *-isomorphic to the image of K , :~ L”(Ea,p) 3 f sp(f)E rU(U)‘defined by (0, I n,(f)r,(A)R,) = J d p ( w ’ ) f ( w ’ ) w ’ ( A ) , and, for A, B E U, K , ~ ( A ) ~ ~ ~ ( = B )7r,(B)P7rU(A)R2,. R, When B = {P}’n3(W*(7r,))= 3(W*(7rU)), or equivalently, ~ ( W * ( ? TC ,)) {P}’, p is called a central measure, for which we can derive the following result from the above fact:

151

Proposition 2 ([16]) A map A, defined b y

A p : 7ru(M)” 3 7ru(A)H

~ , ( aE)3(W*(7ru))

is a conditional expectation characterized b y

To summarize, we have established the following logical connections: 1) As dual of embedding 3 ( W * ( % ) )9 W*(!?l) of the centre, we obtain a basic q+c channel En ++ Prob(Spec(3(W*(%)))= Prob(M) with a factor spectrum M = Fa/ = as the classifying space of sectors. 2 ) A central measure pu with a barycentre w = w’dpu(w’) E Ea specifies a conditional expectation Apw : W*(7ru)3 7ru(A) = [Spec(3(W*(7ru)))3 w’ Hw ’ ( A ) ] E 3(W*(7ru)), whose dual h

h

-~ , ~ ( a ) -

sEB

A;w : Prob(Spec(3(W*(7ru)))

+

Ew*(7rW) h

defines a c+q channel given by S p e c ( 3 ( W * ( 7 r u ) ) ) [MI ~ 3 y wy := AEw(6,) = 6, 0 A, E supp(p,) c Fg[C Ez] as a (local) section of the h

bundle Fa -+ [Fa/ M ] = M. 3 ) Operationally, this corresponds just to a choice of a selection criterion to select out states of relevance and we have realized that the more internal structure to be detected, the larger algebra we need, which requires the Galois extension scheme just in parallel with DHR sector theory and with my propsal of general augmented algebra, as seen below. 2.2

Selection criteria to choose an appropriate family of sectors

Now we come to a “unified scheme for generalized sectors based on selection criteria” [13,14], extracted from a new general formulation of local thermal states in relativistic QFT [2, 121. What I have worked out so far in this direction can be summarized as follows:

[ [

B) DHR sector theory of unbroken internal symmetry: discrete sectors

A) Non-equilibrium local states: continuous sectors

1

C) Sector structure of broken symmetry: discrete & continuous

1:

[

D) Unified scheme for Micro-Macro based on selection criteria

1

1

A) General formulation of non-equilibrium local states in QFT [2, 12, 131; 152

B) Reformulation [14]of DHR-DR sector theory [5,6]of unbroken internal symmetry;

C) Extension of B) to spontaneously or explicitly broken symmetry [14, 151. The results obtained in A), B) and C) naturally lead us t o

D) Unified scheme for describing Micro-Macro relations based on selection criteria [12, 13, 141: i,

[

q : generic states of object system

]

+ll) "

[

c : reference model system with

classifying space of sectors

iii) a map to compare i) with ii)

h

4

state preparation tk selection criterion: ii) a i)

classification & interpretation of i) w.r.t. ii): i) ii) Q-c

c-Q

which can be seen as a natural generalization of

1

1

,

Example 3 The formulation of a manifold M based o n local charts { ( U x ,cpx : Ux + an)} consisting of i)= local neighbourhoods lJx of M constituting a covering M = W x , ii)= model space R", iii)= local homeomorphisms cpx : Ux --t R", iv)= interpretation of the atlas in terms of geometrical invariants such as homology, cohomology, homotopy, K-groups, characteristic classes, etc., etc. Example 4 Non-equilibrium local states in A ) [2, 12, 131 are characterized by localizing the following generalized equilibrium states with fluctuating thermal parameters: i) = the set Ex of states w at a spacetime point x satisfying certain energy bound locally [w((l + Ho)") < 00 with "local Hamiltonian" Ho], ii) = the space BK of thermodynamic parameters (p,p ) to distinguish among different thermodynamic pure phases and the space M + ( B K ) =: T h of probability measures p on BK to describe fluctuations of (p,p ) , iii) = comparison of an unknown state w with members of standard states wp = C * ( p ) = d p ( p , p ) ~ pwith , ~ parameters p belonging to reference system, in terms of the criterion w = C * ( p ) through "quantum fields at x" E I,

sBK

(justified b y energy bound in i). iv) = adjunction

z

153

with q-+c channel as a Yefl adjoint” to the c-+q channel C* (from the classical reference system to generic quantum states): as a localized f o r m of the zeroth law of thermodynamics, this adjunction achieves simultaneously the two goals of identifying generalized equilibrium local states and of giving the thermal interpretation (C*)-’(w) = p of a selected generic state w in C*c?i)

the vocabulary of a standard known object p E Th.

What we have discussed so far can be summarized as follows: 1. Classification of quantum states/representations by quasi-equivalence (= unitary equivalence up to multiplicity): achieved by means of sectors labelled by macroscopic order parameters as points in the spectrum of centre, where a sector is defined by a quasi-equivalence class of factor states w E Fa with trivial centres 3(W*(ru)):= W * ( r un ) W*(ru)‘= C1fiw. In short, a sector = all density-matrix states within a factor representation = a folium of a factor state.

U 2. A mixed phase = non-factor state = non-trivial centre 3(W*(U)) # C1,j: allows “simultaneous diagonalization” as a central decomposition arising from non-trivial sector structure. ===+ 3(W*(U)): the set of all macroscopic order parameters to distinguish among different sectors; S p e c ( 3 ( W * ( % ) ) ) :a classifying space to parametrize sectors completely in the sense that quasi-equivalent sectors correspond to one and the same point and that disjoint sectors to the different points.

U 3. Micro-macro relation: Intersector level controlled by 3 ( W * ( U ) ) :macroscopic situations prevail, which are macroscopically observable and controllable; Inside a sector: microscopic situations prevail (e.g., for a pure state in a sector, as found in the vacuum situations, it represents a “coherent subspace” with superposition principle being valid).

4. Selection criterion = physically and operationally meaningful characterization as to how and which sectors should be picked up for discussing a specific physical domain. E.g., DHR criterion for states w with localizable charges (based upon “Behind-the-Moon” argument) nu r q q g T O ra(o/)in reference to the vacuum representation TO.

A suitably set up criterion determines the associated sector structure so that natural physical interpretations of a theory are provided in a physical domain specified by it.

154

3

Sectors and symmetry: Galois-Fourier duality

To control the relations among algebras with group actons, their extensions and corresponding representations, we need the Galois-Fourier duality as an important variation of our main theme Micro-Macro Duality. The essence of DHR-DR theory [5, 61 of sectors associated with an unbroken internal symmetry can be seen in this duality which enables one to reconstruct a field algebra 5 as a dynmaical system 5 OG with the action of an internal symmetry group G from its fixed-point subalgebra Q = TG consisting of G-invariant observables in combination with data of a family 7 of states E Ea specified by the above DHR selection criterion: Invisible micro Visible macro

In my recent reformulation, its applicability range restricted to unbroken symmetries has been extended to not only spontaneously but also explicitly broken symmetries. In B) DHR-DR sector theory, we see 1. Sector structure:

2. s(T(u)”) = @*C(I,, €3 IV,) -@

=

P ( G ) ; G = s ~ ~ c ( ~ ( T ( Q ) /==+ ’))

vocabulary for interpretation of sectors in terms of G-charges.

3. ( ~ ~ ~sector 4 ~of )U: (y,V,) E G : equiv. class of irred. unitary representations of a compact Lie group G of unbroken internal symmetry of field algebra 5 := U €3 c3d with a Cuntz algebra generated by isometries.

09

4. (T,U,4): covariant irred. vacuum representation of C*-dynamical system

5 .A G, s.t. T ( T ~ ( ( F ) ) = U(g)T(F)U(g)*. 7

5 . U, G, 5: triplet of Galois extension 5 of Q = EG by Galois group G = Gal(S/U), determining one term from two. How to solve two unknowns G & 5 from U?:DHR selection criterion

155

==+

I (C End(%)): DR tensor category

E RepG

TannahKrein

==s

duality

G ==+

grUxG.

Similar schemes hold also for C) with spontaneously and/or explicitly broken symmetries. For instance, in the case of SSB, we have [14] -broken: G IJH : unbroken n

T E! RepH

n

Ud =

II - ~M(HTG)=~~XG

-H

zH

~ - - tIISHEG/HgHg-’

G/H:

1

U

sector bundle

with 3,(Ud) = L“(H\G;dg)@3,(Ud) = Lw(H\G; d g ) @ l w ( A )and the base space G / H of the sector bundle, Spec(3,(%’)) = ugHEc/Hgfig-’ -+ G / H , corresponds mathematically to the “roots” in Galois theory of equations and physically to the degenerate vacua characteristic to SSB.

3.1

Hierarchy of symmetry breaking patterns and augmented algebras

Extension of B) to broken symmetries [14, 151: In my attempts to extend DHR-DR sector theory with unbroken symmetries to the broken cases, the adjunction, augmented algebra

Broken

z

Unbroken,

has been important, as seen in my criterion of symmetry breaking:

Definition 5 ([14])A symmetry described b y a (strongly continous) automorphic G-action r : G n 5(: field algebra), is unbroken in a given repre7

sentation ( ~ , 4 j )of 5 i f the spectrum Spec(3,(3)) of centre 345) := s(5)”n ~ ( 5 )is’ pointwise invariant (p-a.e. w.r.t. the central measure p which decomposes T into factor representations) under the G-action induced on Spec(3,(5)). If the symmetry is not unbroken in ( ~ , 4 j ) )it is said to be broken there.

Remark 6 Since macroscopic order parameters Spec(3,(5)) emerge in lowenergy infrared regions, a symmetry breaking means the “infrared(=Macro) instability” along the direction of G-action. Remark 7 Since a representation 7~ with broken symmetry can still contain unbroken and broken subrepresentations, further decomposition of Spec(3,(5)) is possible into G-invariant domains. A minimal G-invariant domain is characterized by G-ergodicity which means central ergodicity. ==+ ?r is

156

decomposed into a direct sum (or, direct integral) of unbroken factor representations and broken non-factor representations, each component of which is centrally G-ergodic. phase diagram on Spec(3,(8)). Thus the essence of broken symmetry is found in the conflict between factoriality and unitary implementability. In the usual approaches, the former is respected at the expense of the latter. Taking the opposite choice to respect implementability, we encounter a non-trivial centre which provides convenient tools for analyzing sector structure and flexible treatment of macroscopic order parameters to distinguish different sectors. non-trivial centre

i 2 Unbroken2], Namely, the adjunction holds between [Broken controlled by a canonical homotopy 7 fro? [$.A G with non-implementable broken symmetry G in a pure phase] to [Z .A G with unitarily implementecj symmetry G + U ( G ) in a mixed phase with a non-trivial centre], where 8 is an augmented algebra [14] defined by $ := 5 x (H\G), as a crossed product of 8 by the coaction of H\G (: degenerate vacua) arising from the symmetry breaking from G to its unbroken subgroup H . Note here that the above criterion does not touch upon the relation between the symmetry group G and the dynamics of the physical system described by the algebra 8 in relation with spacetime; if the latter is preserved by the former, the breakdown of symmetry G is called spontaneous (SSB for short). Otherwise, it is explicit, associated with some parameter changes involving changes of physical constants appearing in the specification of a physical system. For instance, we can formulate such an explicitly broken symmetry as broken scale invariance associated with temperature as order parameter [15], where augmented algebra of ob-

servables & = !2l x (S0(3)\(B+ x L i ) is the scaling algebra due to Buchholz and Verch [4] to accommodate the notion of renormalization group (in combination with components arising from SSB of Lorentz boost symmetry due to thermal equilibrium [lo] to accommodate relative velocity u!-’ := p!-’/pE S0(3)\Li). What is scaled here is actually Boltzmann constant kB!! In this way, we are led to the hierarchy of symmetry breaking patterns ranging from unbroken symmetries, spontaneous and explicit breakdown of symmetries, the latter of which would be related with more general treatments of transformations, such as semigroups or groupoids. An eminent feature emerging through the hierarchy of symmetry breaking patterns is the phenomena of ezternalization of internal degrees of freedom in the form of order parameters and breaking parameters, along which external degrees of freedom coupled to the system are incorporated through Galois extension into the augmented algebra: it describes a composite system consisting of the microscopic object system and its macroscopic “en*To be precise, “unbroken” should be understood as “unitarily implemented”.

157

vironments” , which canonically emerge at the macroscopic levels consisting of macroscopic order parameters classifying different sectors and of symmetry breaking terms such as mass rn and kg, etc. This formulation allows us to describe the coupling between the system and external fields in a universal way (e.g., measurement couplings).

4

From [thermality 2 geometry] towards [history of Nature]

Although the modular structure of a W*-algebra in standard form has not been explicitly mentioned so far, it plays fundamental roles almost everywhere in the above discussion, responsible for the homotopical extension mechanism: this is crucial, for instance, in the formulation of group duality and of scaling as well as conformal aspects. From the viewpoint that the notion of quasi-equivalence fundamental to our whole discussion is just a form of homotopy, we show here the Galois-theoretical aspects of modular I ,

structure !TI Z arising from canonical homotopy 7, : 7r to standard form.

Theorem 8 ([lS]) i ) I n the universal representation

-+

rooto move

(7ru,fju

=

@

3,)

WEE%

of a C*-algebra %, we define the maximal representation no disjoint from a representation 7r = (.,fir) E R e p a b y 7r’ := sup{p E

Repa; p

5 xu,p

7r).

Then we have the following relations in terms of the projection P(7r) E W*(%)’on the representation space sj, of rr and its central support c ( n ) : ,

7r1

I

5 7r2 ===+ 7r; 2 7r;,

P(7r’) = . I ) (

7ro

, , I

= noooand

7r

5 roo,

:= 1 - c ( n ) ,

, I

P(7r””)= .(.)I1

= c(7r) =

v

UEU(?F@)’)

ii) Quasi-equivalence n1 = 7r2(* 7rl(%)” W*(7r1)*= W * ( q ) *is ) equivalent to 7rY0

uP,u* E P(3(W*(8))). N 7r2(8)”

c(r1) = c ( ~ 2 )

= 7rg0.

( 1

, I

iii) The representation (7roo,c(7r)fiu)of W*-algebra W*(7r)N

7roo(%)”

in

the Hilbert space c(n)fiu = P(7rr”’)fi,gives the standard f o m of W * ( r ) associated with a normal faithful semifinite weight p and the corresponding 158

Tomita- Takesaki modular structure (J,,,,A,,,). It is characterized b y the universality: Std(7roo,a ) N Repn(.rr,a), where Std denotes the caterogy of representations of Q in standard form; according to this relation, any intertwiner T : 7r -+ a to a representation I ,

(a,&,) in standard f o r m of W * ( a )is uniquely factored T = Toooqr through

the canonical homotopy qT : T \ \

,,

3

r o owith a uniquely determined intertwiner

Too: roo+ a. iv) The quasi-equivalence relation 7r1 M 7r2 defines a classifying groupoid F a consisting of invertible intertwiners in the catego y R e p n of representations of U, which reduces on each T E R e p a to Fa(7r,7r) N Isom(W*(7r),), the group of isometric isomorphisms of predual W*(7r),as a Banach space. The modular structure in iii) of W*-algebra W * ( T )=: M in the standard $ 8

form in (7roo,c(7r)fj,) can be understood as the minimal implemention b y the unitary group U(M’) of a normal subgroup Gm := I s o m ( M * ) m Q I s o m ( M , ) fiing M pointwise: namely, for y E Gml there exists U; E U(M’) s.t.

(yw,x)= (w,y*(x)) = (w,U?xU;)f o r w

E

M,

,

and UyxU!, = x Jx E M. For M of type 1111we can verify Galois-type relations involving crossed product b y a coaction of the group Gm N U(m’) as follows: = MV

=MM

M = (MV

G:Galois extension of M,

; fized-point

subalgebra under Gm,

Gm = Gal(3(M)’/M): Galois group of M ~f

according to which factoriality 3(M) = C 1 of M can be seen as the ergodicity ofm under Aut(!Bl) or Gm:

CI = Mn

=

nu(M’)/ = ( m m l ) G m2

(m’)Aut(m).

In view of the dominant roles of thermal or modular-theoretical notions mentioned above, this theorem suggests possible paths from thermality to geometry t o explain different geometries at macroscopic classical levels emerging from the invisible microscopic quantum world; it would explain the origin of universality of Macro put in Micro-Macro Duality in our theoretical descriptions of physical worlds. A typical example of this sort can be seen in the formulation of group duality which exhibits its essence as a homotopical duality involving interpolation spaces IS]. Moreover, we can develop a framework t o go into a step from the above modular homotopy t o the generalized version of classifying spaces or classifying toposes [16]. Along

159

this line of thoughts, we can envisage such a perspective that theoretical descriptions of physical nature can be mapped into a “categorical bundle of physical theories” over a base category consisting of selection criteria to characterize each theory as a fibre, which are mutually connected by metamorphisms of intertheory deformation arrows parametrized by fundamental physical constants like ti, c, Ice; n, e , etc., controlled by the “method of variations of natural constants” (work in progress). One of the most important virtues of the above augmented algebra is found in the possibility that such physical constants can be treated on the same footing as various physical variables responsible for changing the symmetry properties of the systems; in such contexts, they represent controlling parameters of deformations among different selection criteria to determine theories corresponding t o stabilized hierarchical domains. Then the most crucial step will be to formulate each selection criterion as an integrability condition in terms of generalized categorical connections, through which the framework can accommodate such an adjunction as

[

irreversible historical process

I =

[

homotopical

stabilized hierarchical domains with reversible dynamics

1

to be found among such adjunctions as to put a generic category with noninvertible arrows (describing an irreversible open system in a historical process) in a relation adjoint to a groupoid with invertible arrows (corresponding to a reversible closed system with repeatable dynamics in a specific hierarchical domain). This kind of theoretical framework would provide an appropriate stage on which the natural history of cosmic evolution be developed.

References [l] Baaj, S. and G. Skandalis, Ann. Scient. Ecole Norm. Sup. 26 (1993), 425-488.

[2] Buchholz, D., Ojima, I. and ROOS,H., Ann. Phys. (N.Y.) 297 (2002), 219 - 242.

[3] Bratteli, 0. and Robinson, D.W., Operator Algebras and Statistical Mechanics, vol. 1, Springer-Verlag (1979). [4] Buchholz, D. and Verch, R., Rev. Math. Phys. 7 (1995), 1195-1240. [5] Doplicher, S., Haag, R. and Roberts, J.E., Comm. Math. Phys. 13 (1969), 1-23; 15 (1969), 173-200; 23 (1971), 199-230; 35 (1974), 49-85. [6] Doplicher, S. and Roberts, J.E., Comm. Math. Phys. 131 (1990), 51107; Ann. Math. 130 (1989), 75-119; Inventiones Math. 98 (1989), 157-218. 160

[7] Enock, M. and Schwartz, J.-M., Kac Algebras and Duality of Locally Compact Groups, Springer, 1992. [8] Maumary, S. and Ojima, I., in preparation. [9] Nakagami, Y. and Takesaki, M., Lec. Notes in Math. 731, Springer, 1979.

[fO] Ojima, I., Lett. Math. Phys. 11 (1986), 73-80. [ll]Ojima, I., J. Stat. Phys. 56(1989), 203-226; Lec. Notes in Phys. 378, pp.164178, Springer, 1991. [12] Ojima, I., pp. 48-67 in Proc. of Japan-Italy Joint Workshop on F’undamental Problems in Quantum Physics, Sep. 2001, eds. Accardi, L. and Tasaki, S., World Scientific (2003). [13] Ojima, I., pp.365-384 in “A Garden of Quanta”, World Scientific (2003); e-print: cond-mat/0302283.

[14] Ojima, I., Open Systems and Information Dynamics, 10 (2003), 235279; math-ph/0303009. 1151 Ojima, I., Publ. RIMS 40, 731-756 (2004). [16] Ojima, I., in preparation. [17] Ozawa, M., J. Math. Phys. 25, 79-87 (1984); Publ. RIMS, Kyoto Univ. 21, 279-295 (1985); Ann. Phys. (N.Y.) 259, 121-137 (1997). [18] Ozawa, M., quant-phys/0310072, to appear in Phys. Lett. A.

161

White noise measures associated to the solutions of stochastic differential equations Habib Ouerdiane University of Tunis El Manar Faculty of Sciences of Tunis. Campus universitaire, 1060 Tunis.Tunisia, E-mail: [email protected]

1

Introduction

Let N be a complex Frkchet nuclear space with topology given by an increasing family of Hilbertian norms (1 . n E N}. It is well known that N may be represented as N = nnEWN, where the Hilbert space N, is the completion of N with respect to .1, By the general duality theory N' is given by N' = UnE&, = NA is the topological dual of N,. Let 8 : R+ + R+ be a where continuous convex strictly increasing function such that

In,

lim -= 03, 2-00

z

O(0) = 0.

Such functions are called Young functions. For a Young function 8 we define

o * ( ~ )= sup(tz - e ( t ) ) , t2o

(2)

This is called the polar function associated to 8. It is known that 8* is again a Young function and (0.). = 8. For every p E Z and m > 0, we denote by Ezp(p(Np, 8, m) the space of entire functions f on the complex Hilbert space NP such that lIfll~,p,m := SUP If(z)le-e(m'zlp)< +03. (3) Z€Np

We fm a Young function 8. Then {F',m(N-p):= Ezp(N-,, 8, m ) ; p E N,m > O} becomes a projective system of Banach spaces and we put

Fe(N') = proj lim Ezp(N-,, 8, m) p+w;m+O

162

(4)

which is called the space of entire functions on N’ with an 8-exponential growth of minimal type. On the other hand {Ezp(N,, 8, m ) ;p E N,m > 0} becomes an inductive system of Banach spaces and we put

This is called the space of entire functions on N with 8-exponential growth of arbitrary type. Then .Fo(N’)equipped with the projective limit topology is our test function space. The corresponding topological dual, equipped with the inductive limit topology, is denoted by 3z(N’)which is the generalized functions space, see [8] for more details. In particular, if N = &(R) (the complexified of the Schwartz test function space S(R)) and 8(z) = z2, then .Fo((N’) is nothing than the analytic version of the Kubo-Takenaka test functions space and the corresponding topological dual is the Hida distributions space, see [9]. The test functions space of Kondratiev-Streit type ( S ) p , /3 E [0,1) are obtained choosing 8(z) = z h , see [14], [15], [21], [23]. More recently, it was introduced a two-variable version of the above spaces, see [ll].In fact for arbitrary k E N, we can replace the nuclear space N by the product

NI x . . . X N k , and 8 by (81,. . . 8 k ) where 8i are Young functions and n/i is a complex nuclear Frbchet space, 1 5 i 5 k . Then it is possible to extend all the results obtained in [8] in the mulivariable case. In particular, the Laplace transform L: induces the following topological isomorphism

F;(N{ x . . . x N;) Gfj*(Nlx * . . x N k ) (Nl x . . .x N ~is )the space of entire functions on N1 x . . .x Nk with

and where 8*-exponential growth of arbitrary type with respect to 8* = (8;, . . . , 8 9 , where 8; is the polar function corresponding to &. Another important result in [5] and [6]is the characterization theorem for convergent sequences of distributions in .F;(Ni x . . . x NL). Using this result, we can directly define for any given continuous stochastic process X ( t ) E Fi(N; x . . . x N;) the integral

Very useful in applications is the convolution product on 3;(N’),see [4], and [6] for details. In fact, we define the convolution of two distributions @, Q E Fi(N’) bY @ * Q = c-l(c@. cq (7) which is well defined because Go* ( N )is an algebra under pointwise multiplication. We can define for any generalized function @ E Fi(N’) the convolution exponential of @ denoted by exp* @ as a generalized function on 3;*,@. ). (N’) Note that for a generalized function E ( S ) &the Wick exponential of @ denoted by expo @ does not belong to ( S ) &but , it belongs to a bigger space of 163

distributions (S)-’ called Kondratiev distribution space, see [13]. In this work, we do not restrict ourselves to the theory of Gaussian (White noise) and non-Gaussian analysis studied for example in [2], [9],[lo], [13], [14] and [15] but we develop a general infinite dimensional analysis. First, we give a decomposition of convolution operators from Fe(N’) into itself, into a sum of holomorphic derivation operators. Then, we establish a topological isomorphism between the space L(Fe(N’),Fe(N’))of operators and the space Fe(N’)&p (N) of holomorphic functions. and we Next, we develop a new convolution calculus over L(Fg(N’),Fe(N’)) give a sense to the expression eT := &o for some class of operators T . As an application of this theory we solve some linear quantum stochastic differential equations. Finally using a recent result obtained in [18]and concerning white noise measures satisfying an exponential decay property, we give a asymptotic estimates of solutions of stochastic differential equations.

5

2

Preliminaries

For any n E M we denote by N On the n-th symmetric tensor product of N equipped with the 7r-topology and by N:” the n-th symmetric Hilbertian tensor product of Np. We will preserve the notation [.Ip and I. l -p for the norms on NF”and N?; respectively. We denote by (., .) the C-bilinear form on N’xN connected to the inner product (.I.) of H =NO, i.e. ( z , t )= (ZIE) , z E H , t E N. By definition f E Fe(N’) and g E Be(N) admit the Taylor expansions: m

f(z)

=

g(E)

=

C(ZBn,fn),

n=O

ZEN’,

fn

EN’’’

(8)

00

C(gn,Pn),

F E N , gn

E

(Nan)’

n=O

where we used the common symbol (., .) for the canonical bilinear form on (No”)’x Nan for all n. In order to characterize Fe(N’) and Be(N) in terms of the Taylor expansions, we introduce weighted Fock spaces Fe,m(Np)and G s , m ( N - p ) .First we define a sequence (0,) by

Suppose a pair p E we put

4

Then, for f = (fn)F.owith fn E N p

N,m > 0 is given. 00

n=O

m

n=O

164

Accordingly, we put

Finally, we define

Fe(N)= p--too;mlO projlim Fe,,(Np),and

Go(”)

=

indlim Ge,,(N-,).

p-+w;m--tw

(10)

It is easily verified that Fe(N) becomes a nuclear RCchet space. By definition,

Fe(N) and Ge(N’) are dual each other, namely, the strong dual of identified with Go (N’) through the canonical bilinear form:

Fe(N) is

n=O

The Taylor series map T (at zero) associates to any entire function the sequence of coefficients. For example, if the Taylor expansion o f f E Fo(N’) is given as + in (8), the Taylor series map is defined by If= f = (fn). In particular, for every z E N‘, the Dirac mass 6, defined by << 6,,(p >:= p(z) , belongs to Fi(N’). Moreover, b, coincide with the distribution associated to the formal -+ 8series 6, := ( ~ ) , Q O .

Theorem 1 [8] The Taylor series map T gives two topological isomorphisms Fe(N’)

3

-+

Fe(N), Be*(N) + Ge(N’).

Application to white noise analysis

For some functions 8,the spaces Fe(N’) and Be(N) play an important role in the theory of Gaussian and non Gaussian analysis. In fact let X c H c X’ be a real F’rkchet nuclear triplet. Let y be the standard Gaussian measure on (XI, B) where B is the a-Borelian algebra on X’,determined via the BochnerMinlos theorem by the characteristic function:

e ) ~

F.

and llElli = (6, is the Hilbertian norm in the space By complexification of the real triplet X c H c X’we obtain N c 2 c N where N = X iX and 2 = H + iH. Suppose that lim < 03 . Then Fe(N’) can by densely topologically embedded in the Hilbert space L2(X’,y) and we can construct the following Gelfand Triplet Fe(N’)

c L 2 ( X ’ , y )c FG(N’) 165

+

(13)

3.1

S-Transform

Let 0 be a Young function. Denote by Fi(N’) the strong dual of the test functions space Fe(N’). From condition (1) we deduces that for every E E N , the exponential function ec defined by e c ( z ) = e(’,t),z E N’ belongs to the space Fe(N’). The Laplace transform L of a distribution 4 E Fi(N’) is defined by L(4)(I)= &I)= ((4,ec)), I E N . (14) By composition of the Taylor series map with the Laplace transfoLm, we deduce that 4 E .Fi(N’) if and only if there exists a unique formal series 4 = (q5n)n20 E Ge ( N )such that ?(I)= Cn>O(E*n, &). Then, the action of the distribution 4 on a test function cp(z) = is given by

c

<< 4, ‘p x=-= n!(4n,‘pn) .

(15)

n>O

In the white noise Analysis we use the S-transform

1 S(4)(E):= L 4 ( 0 exp(-,E2),I

E N , $E Fe(N’).

(16)

Let now k , be given nuclear gaussian spaces ( X j c Hj c X$,y)and 0 = (el,&,...,0,) be a multivariable Young function, i.e., &,&, ...,BI, are k given Young functions and denote by

+

+

where Nj = X j iXj Zj = H j iHj. Setting yBk = y @ y ... @ y the k-fold tensor product of the standard gaussian measure. The next result give a characterization of new Gelfand triplet.

Theorem 2 I f we suppose’ that limz+a, < 00 for every 1 5 j 5 k , then Fo(N’) can be densely topologically embedded in the space L 2 ( X ’ , y B k ) and we can construct the following Gelfand triplet: Fe((N’) c L 2 ( X ’ , y B k )c .Fi(N’). Moreover the chaotic transform (S-Transform) realizes a topological isomorphism of nuclear triplets :

c L 2 ( X ’ , y B k ) c 3:(N’) 1s 1Is Fe(N’) c Fock(2’”) c Be*(N) To(”)

1

where I s is the Wiener -ItB-Segal isometry and F o c k ( Z k ) is the Bosonic Fock space on Zk and e* = (el, e2,...,&)* = (e;, e;, ...,e;).

166

3.2

Relation of this theorem with previous results

1. If k

=

1 we obtain the results of [8]. In particular if e(z) = $ , a > 1

+

with 1 - 1 and we obtain in this case the usual then @*(z)= a a space of entire functions of exponential type, see e.g., [21], [22] and [23]. For every Fe(N) we have

f~

If a = 2 and X is the Schwartz space S(R),the space Fj,(S(R)) is the Hida distributions space, see [lo] and [9]. 2. The Potthoff-Streit characterization theorem, see [24], is a particular case of the general topological isomorphism: F:(N‘) + 90- ( N )where k = 1, O ( t ) = t2 and X = S(R). 3. In the particular case where k = 1, and N is a arbitrarily Banach complex space B and e(t) = t a ,a 1 1 the spaces Fe(N’),Fe(N),L&(N),G,(N’) are introduced first by the author in [20], and the analog of Theorem 1 is given in this case. 4. In [7] Cochran-Kuo-Sengupta introduce the CKS space of distributions [v]: where a = ( C Y , ) , ~ N is a positive sequence and G,(t) = - a(n)f is a n analytic function. If we put 8 * ( t ) = Log(G,(t2)) then [v]: = Fi(N). The hypothesis of the analycity of the function G,(t) in [7] is not necessary in our case, moreover we here obtain explicitly the space test functions and also a characterization theorem for this space.

En>,,

4

Convolution calculus

In the next we develop a new convolution calculus over generalized functionals space .Fi(N’). Unlike the Wick calculus studied by many authors, see [9], [15], [16], [14] and [23], the convolution calculus is developed independently of the Gaussian Analysis. In fact for 4 E Fi(N‘) and ‘p E Fe(N‘) the convolution of 4 and cp is defined by

( 4 * Cp)(Z) :=<<4, T - z p >> ,

2 6

N’

(17)

where T-, is the translation operator, i.e., T-,(P(z) = cp(z + z), z E N’ and for into itself. A every t E N’, the linear operator T-= is continuous from Fo(N’) direct calculation shows that 4 * ‘p E Fo(N’). Let 4 1 , 4 2 E Fi(N‘),we define the convolution product of 41 and 4 2 , denoted by 41 * 4 2 by

<< 41 * 4 2 , cp >> := [4i* ( 4 2 * cp)](O) , cp E .Fe(N’)

167

4.1

Convolution operators

In infinite dimensional complex analysis, a convolution operator on the test space Fe(N') denoted for simplicity by Fe is a continuous linear operator from Fe into itself which commutes with translation operators. It was proved in [4] that T is a convolution operator on Fe if and only if there exists 4~ E F; such that Tcp = 4~ * cp , V cp E 3 e . (18) Moreover, if the distribution 4~ is given by CnzO(z@n, p n ) E Fe then

&-= (4m)m>o E Go

and p(z) =

dm and cpm+n

of order m ,

where (q5m, cpm+n)m denotes the right contraction of see [15]. In particular, we have

T(eE)(z)= dT * eE(z) = &<)eE(z). Let 0 be a Young function, y E N' and p(z) = Cn,O(z@n,cpn) E 3 0 , then we define the holomorphic derivative of cp at the point F E N' in a direction y by ~ v c ~ (:= z)

C(n+ l ) ( z m n ,(y,cpn+l)l).

n20

For each m E N the m-linear operator D : N'x ... x N' bY (YI,...,Ym) DyI...Dyrn

-

-

C(F0,3 0 ) defined

is symmetric and continuous, hence it can be continuously extended to N'Dm, i.e., D : &, E dam H D+- E L(Fe,Fe) The action of the operator Dbrnon a test function cp(z) = CnlO(~@n,cpn) is given by (n+ m)! Z@n Db,(cp)(Z) = 7 ( (4ml Pn+m)m). (20)

c

9

n>O

Then, in view of (18), (19) and (20), we give an expansion of convolution operators in terms of holomorphic derivation operators.

Proposition 3 Let T E C ( 3 ~ , 3 ~ Then ) . T is a convolution operator if and only if there exists = ($m)m20 E GO such that

4

m>O

168

Let T+= Cm20 D+,,, be a convolution operator and n E W. Then equality (18) shows that T$ := Tb o ... o Tb = Tp-. In particular n

4.2

Symbols of operators

We denote by L(30,3 8 ) the space of continuous linear operators from 3 0 into itself, equipped with the topology of bounded convergence. In this subsection we define the symbol map on the space L(30,30). Then we give an expansion of such operators in terms of multiplication and derivation operators.

Definition 4 Let T E L(30,F0), the symbol a(T) of the operator T is a complex -valued function defined by a(T)(z,E ) := e-(*gE)T(eC)(z),

z E

N',

E

N.

-

Similar definitions of symbols have been introduced in various contexts, see [ll], [12], [15], [16], and [22]. In the general theory, if we take two nuclear F'rQchet spaces X and V then the canonical correspondence T KT given by

(Tu,w) = ( K T , u @ v ) ,u E X,WE V', yields a topological isomorphism between the spaces L ( X , V ) and X'GV. In particular if we take X = V = 3 0 which is a nuclear FrQchetspace, then we get L ( 3 0 , 3 0 ) F,G30. So, the symbol u ( T )of an operator T can be regarded as the Laplace transform of the kernel KT

a ( T ) ( z , ( )= K T ( e c @ 6,)

,

z E

N' , E

E

N.

(21)

Moreover, with the help of equality (21) and theorem 2, we obtain the fallowing theorem. Theorem 5 The symbol map yields a topological isomorphism

L(30,Fe)-+ FeGGi.

-

More precisely, we have the following isomorphisms

.C(F~,F~)

T H a(~)(z,t) =

C

FeGGi

S.T

(Kl,m, 2 ~ B' t ~ m H )

F~GG~

2 = (Kl,m)l,m>o.

l,m>0

Example 6 1) The symbol of a convolution operator T+= bu

I69

-

D+- is given

2) If we denote by M f the multiplication operator by the test function f . Its symbol is given by

u(Mf)(z,E) = e - ( " p E ) ( f e c ) (= z )e - ( " y c ) f ( z ) e c ( z=) f ( z ) . Let 2 E F&Ge and assume that -+ operator T associated to K satisfies

T

2 = f@ $ = (fl

@ 4 m ) l , m > ~Then .

= MfT4,

the

(22)

where f (2) = C120(z@'1, f l ) and T4 is the convolution operator associated to the distribution 4 given by $. Moreover, we have

T

= MfT+= a ( M f ) ( zD)a(T+)(z, , D ) = u ( T ) ( zD, ) .

Thus, using the density of Fe @ Ge in F&Ge, we obtain the following result.

Proposition 7 The vector space generated by operators of type (22) is dense in C(30,Fe).

4.3

Convolution product of operators

Let T1,T2 two operators in C(Fe,Fe),the convolution product of TI and T2, denoted by TI * T2, is uniquely determined by a(T1 * T2) = a(Tl)u(T2).If the operators TI and T2 are of type (22), i.e., TI = MflT+,and T2 = Mf,T+, then

TI * T2 = MflfiT',*cp,. In particular, if T = MfT+then for every n E N we have T*" = MfnT4.n. Let T4 (resp. M f ) be a convolution (resp. multiplication) operator. Then for every n EN TQIfn = T4.n = TT and M;" = M f n = Mf".

Proposition 8 Let T E Lo; then the operator e*T :=

belongs t o

L(F(.O*)*F e e ) . 1

E C(.Fe,F') and put S, = EL=,$. Then using the Laplace transform isomorphism we shows that a(&) converges in Fee@G,e* to eU(*), from which the assertion follows. H Let T E L(F9,Fe) and consider the linear differential equation

Proof. Let T

A

dE

= T E , E(0) = I. dt Then the solution is given informally by : E ( t ) = etT, t E R. In the particular case, where T is a convolution or a multiplication operator; the solution E ( t ) = etT is well defined since eT = erT. If T is not a convolution or a multiplication operator then the following theorem gives a sufficient condition on T to insure the existence of its exponential eT.

170

+

Theorem 9 Let K = (Kl,,) E FeGG,q satisfying (Kl,m,K ~ t , ~ ,=) k0 f o r every m, I' 2 1, m', 1 2 0 and 1 5 k 5 m A 1' and denote by T the operator associated to T h e n T" = T*", V n E N.Moreover

d.

eT = e*T E L(F(,~* ). ,~

~

0

)

.

Proof. Using the last proposition, it will be sufficient to assume that K L ,=~ ( f i @ dm)) i.e., T = MfT4 = MflDh,

c

l,m>O

where f i ( z ) = (z@',f i ) Assume that f i = q@', q E Then it is easy to see that

DLMfl

= MflD4-

+

N

and

$m = y B m ,

y E N'

c

k!CEeC~(Y,77)kMfi-hD4,-b, k=l an equality on Fe. The assumption (&,m, & ' , m ' ) k = 0 implies that ( y , q ) = 0. Then

D4-Mfr

= MflDbm*

Thus, using the density of the vector space generated by {q@l,q E N } in the space N Oland the density of the vector space generated by {y@", y E N ' } in " O m , we can extend the last equality to every f i E N O1 and dm E N'O" such that (&, f i ) k = 0 , V 1 5 k 5 I Am. Hence, we obtain

MfT4 =

c

MflD4-

=

l,m>O

Then we have for every n E

T"

5

DdmMfl = T 4 M f .

M

= (MfT4)" = ( Mf ) "

This completes the proof.

c

l,m>O

(T+)"= M f n T p n = T*".

H

Applications to stochastic differential equat ions

-

A one parameter quantum stochastic process with values in L(F0, Fe) is a family of operators {Et, t E [O,T]}c L(Fe,Fe) such that the map t Et is continuous. Theorem 10 Let t E [O,T]H f ( t ) E Fe and t E [O,T]H d ( t ) E F; be two continuous processes and set Lt = Mf(t)T4(t)T h e n the linear differential equation

'2

_- Mf(t)EtT+(t) I

Eo

has a unique solution Et E L ( ~ ( , )o* ,*Fe,s)given by

Et

= .*(So"

171

Lads),

=I

(23)

Proof. Applying the symbol map to equation (23) t o get - a(Lt)u(Et), a(1) = 1.

--

dt

Then a ( E t )= e-fJ‘(Ls)ds which is equivalent to Et = e*(-fJLsds). Finally, we conclude by the last proposition that Et E t(F(,e*). ,Fee). Theorem 11 Let Lt be a quantum stochastic process with values i n t(Fe,Fe) such that a ( L t ~ s d s )( t , ~=)l ~ ~ ( K i . m ( t ) , 2 ~ 1 W ~ ~ m )

and assume that f o r every t E [O,T],m’,l 2 0 and m,l‘ 2 1 we have

(Ki,m(t), Kp,mi(t))k= 0 , V 1 5 k 5 m A 1’. Then the following differential equation dE = L t E , E(0) = I , dt

has a unique solution in .C(F(,~*)*,F,e) given b y E ( t ) = esi

5.1

Asymptotic estimates for white noise measures

In the sequel we take N = X + i X the complexification of a nuclear Fkechet space X. Let FO(N’)+ denote the cone of positive test functions, i.e. f E FQ(N’)+if f ( y i0) 2 0 for all y in the topological dual X’of X.

+

Definition 12 The space 3O(N’); of positive distributions is defined as the space of 4 E FO(N’)* such that

(4,f ) L 0,

f

E

WW+.

We recall the following results on the representation of positive distributions.

Theorem 13 /19] Let 4 E Fo(N’):. There exists a unique Radon measure pb on X‘, such that

4 ( f )=

1

X‘

f

(y

+iO)d~+(y)~ f

E

Fe(N’).

Conversely let p be a finite, positive Bore1 measure on X’. Then p represents a positive distribution in Fs(N’); i f and only i f p is supported by some X - p , p E W,and there exists some m > 0 such that ee(mlYl-P)dp(y)< 00.

172

Given E E X and x E R, let At," = {y E in X' associated to 6, x.

X'

: ( y ,E )

> x } , denote the half-plane

Theorem 14 Let 4 E Fe(N')* such that r$ defines a (positive) Radon measure p~d,o n X'. For all E E X and x > 0 there exists m > 0 and p E N* such that:

where

c = t1311e,m,p 4

Proof. Using propriety of the Laplace transform of following growth condition

I ~ ( < )5I Cee'(mlElp),

E

E .Fe(N')* we have the

X,

for some m > 0 and p E *. For all t 2 0 we have the Chernoff type inequality:

hence

p$(Ac,=) 5 Ce-(t"-e*(mtl~I~)),t 2 0.

Minimizing in t 1 0 we get, since (8.).

= 8:

From Theorem 13, the result of Theorem 14 holds in particular for all positive distributions r$ E FO(N'):. Applying Theorem 13 and Theorem 14 we obtain a deviation result under an exponential integrability assumption, see [18].

Corollary 15 Let p be afinite, positive Bore1 measure o n X' supported by some X - p , p E N * . Assume that for some m > 0 ,

L-p

ee("'Y'-P)dp(y) < m.

T h e n for all

E X we have:

173

5.2

Tail estimates for solutions of stochastic differential equations

Let c j : [O,T]-+ Fe(N’)* and M : [O,T]-+ Fe(N’)* be two continuous generalized processes, and consider the initial value problem d X-t -

dt

dt * X t + Mt,

X o E Fo(N’)*.

(24)

In the particular case where dt = ado, cy E R, X = S(R) and (Mt)tE(o,T] is a Gaussian white noise on [0,TI, (24) is a classical Omstein-Uhlenbeck equation.

Theorem 16 ([4)The stochastic differential equation (24) has a unique s o h tion in F(,v-l). ( N ) * ,given by X t = X o * e* From the relation

(41

* 42,l)

9ads

+

I‘ .rat e*

+udu

* MJs.

= ( $ 1 , 1 ) ( 4 2 , l),the expectation of

X t satisfies

1fdt,Mt E F ~ ~ ( N forallt ’ ) $ ER+,t h e n e * S ~ ~ s d s a n d e * ~ ~ 9 s Ed sFo;(N’): *Mt and we have the following corollary of Theorem 14 and Theorem 16.

Corollary 17 Let Q; be such that Q;(r) 5 (e“ - l ) * ( for ~ )all r large enough, t > 0. Then the solution X t of (24) and assume that X o , $ t , Mt E Fo;(N’):, belongs to 3 0 ; (N’): and the associated Radon measure (denoted b y p ~ satis~ ) fies

f o r some Ct,rnt,pt > 0 , t E R+

References [l]L. Accardi a n d M. Bozejko:Interacting Fock spaces and gaussianization of probability measures. Infinite Dimensional Analysis, Quantum Probabilty and Related Topics, Vol. 1, no. 4 (1998), 663-670.

[2] S. A l b e v e r i o , Yu. L. D a l e t s k y , Yu. G . K o n d r a t i e v a n d L. S t r e i t :Non-Gaussian infinite dimensional Analysis. J. Funct. Analysis. 138, (1996), 311-350.

[3] N. A s a i , I. Kubo a n d H-H. Kuo:General characterization theorems and intrinsic topologies in white noise analysis, Hirochima Math. Journal, 31 (2001), 229-330.

174

[4] M. Ben C h r o u d a , M. El O u e d a n d H . O u e r d i a n e : Convolution Calculus and Applications to Stochastic Differential Equations. Soochow Journal of Math . 28(4):375-388, 2002. [5] M. Ben C h r o u d a , M. El O u e d a n d H . O u e r d i a n e : Quantum stochastic processes and Applications, to appear. [6] M. Ben C h r o u d a a n d H . 0uerdiane:Algebms of Operators on holomorphic functions and Applications. Math. Physics, Analysis and Geometry, 5 (ZOOZ), 65- 76.

[7] W . G. C o c h r a n , H-H. Kuo and A. S e n g u p t a : A new class of white noise generalized functions, Infinite Dimensional Analysis, Quantum Probabilty and Related Topics, Vol.1, (1998), 43-67. [8] R. Gannoun, R. Hachaichi, H. O u e r d i a n e a n d A. R e z g u i : Un the'ordme de dualite' entre espace de fonctions holomorphes ci croissance exponentielle. J. Funct. Analysis, Vol. 171, No. 1, (ZOOO), 1-14.

[9] T. H i d a , H.-H. K u o , J . Potthof a n d L. S t r e i t : White noise, an infinite-dimensional calculus, Kluwer Academic Publishers Group, Dordrecht, 1993.

[lo] T. Hida:Brownian Motion, Berlin-Heidelberg-New York, Springer Verlag, (1980).

[ll] U. C . J i , N. Obata a n d H . 0uerdiane:Analytic characterization of generalized Fock space operators as two-variable entire functions with growth conditions, Infinite Dimensional Analysis, Quantum Probabilty and Related Topics, Vol. 5, No. 3 (ZOOZ), 395-407. [12] P. K r 6 e a n d R. Raczka:Kernels and symbols of operators in quantum field theory. Ann. I. H. P. Section A , Vol. 18, No 1, (1978), 41-73. [13] Yu. G. K o n d r a t i e v , L. S t r e i t , W. W e s t e r k a m p a n d J.-A. Yan: Generalized functions in infinite dimensional analysis. Hiroshima Math. Journal, 28 (1998), 213-260. [14] H.-H. Kuo: White noise distribution theory, CRC Press, Boca Raton, New York, London and Tokyo, 1996. [15] N. Obata: White noise calculus and Fock space. Lect. Notes in Math. Vol.1577, Springer Vedag (1994). (161 N. Obata: Wick product of white noise operators and quantum stochastic differential equations. J. Math. SOC.Japan Vol. 51, No. 3, (1999), 613-641. [17] N. 0bata:Multivariable White Noise Functions: Standard Setup Revisited, Publ. R.I.M.S, Kokyuroku 1227 (ZOOl), 184-198.

175

[IS] H. Ouerdiane and N. Privault: Asymtotic estimates for white noise distributions. C. R. Acad. Sci. Paris, Ser. I 3 3 8 (2004), 799-804 [19] H. Ouerdiane and A. Rezgui:Reprksentation intkgrale de fonctionnelles analytiques positives. Canadian Mathematical Proceedings( ZOOO), 28283290 [20] H. Ouerdiane: Dualite' et opkrateurs de convolution dans certains espaces de fonctions entieres nucleaires c i croissance exponentielle. Abhandlungen aus der Math. Seminar Hamburg, Band 54 (1983), 276-283. [21] H. Ouerdiane: Fonctionnelles analytiques avec condition de croissance et applications ri l'analyse gaussienne. Japanese Journal of Math. Vol. 20, No.1, (1994), 187-198. [22] H. Ouerdiane: Noyaux et symboles d'opkrateurs sur des fonctionnelles analytiques gaussiennes. Japanese Journal of Math. Vol 21. No.1, (1995), 223-234 . [23] H. Ouerdiane: Algkbres nuclkaires de fonctions entikres et equations aux derivkes partielles stochastiques. Nagoya Math. Journal. Vol. 151, (1998), 107-127. [24] J. Potthoff and L. Streit: A characterization of Hida Distributions, J. Funct. Analysis, Vol. 101 (1991), 212-229.

176

A REMARK ON SETS IN INFINITE DIMENSIONAL SPACES WITH FULL OR ZERO CAPACITY JIAGANG REN School of Mathematics and Computational Science, Zhongshan University, Guangzhou, Guangdong 510275, P.R. China

MICHAEL ROCKNER Fakultiit fur Mathematik, Universitiit Bielefeld, 3361 5 Bielefeld, Germany

Abstract We give a simple proof that for classical Dirichlet forms on infinite dimensional linear state spaces the intrinsic closure of a set of full measure has full capacity. Furthermore, we show that the C,,,-capacity of a set, enlarged by adding the linear span of a basis in the generalized CameronMartin space remains zero if it was zero for slightly bigger capacities a priori.

1.

INTRODUCTION, FRAMEWORK AND A RESULT ON SETS WITH FULL CAPACITY

In infinite dimensional analysis the question whether a given set has zero or full capacity (in the sense that its compliment has zero capacity) is much less studied than in finite dimensions. This question is of importance, since roughly speaking capacity zero sets are not hit by the underlying process whereas a set of full capacity carries the process for all times. The first aim of this paper is to give a simple analytic proof for the fact that the intrinsic closure of a set of full measure has full capacity (cf. Theorem 1.4 below). This fact is essentially known to experts. We refer e.g. to [7] where this result was proved for a class of Dirichlet forms with non-flat underlying state space. But there is no reference for this result for general classical Dirichlet forms of gradient type on linear state spaces. In this case there is quite an easy proof which we present below. The second aim of this paper is to prove a result one would expect, but appears to be new. Namely, we prove that the Cl,,-capacity of a set, enlarged by adding all finite linear combinations of a basis in the generalized Cameron-Martin space, remains zero if it was zero for (slightly bigger) C,,,-capacities, r > 1, p > q, a priori (cf. Theorem 3.3 below). Let us first describe our framework, in which we strictly follow [2]. Let E be a separable Banach space over R.Let E' denote its dual and B ( E ) its Bore1 u-algebra. Let ( H ,(,)) be a Hilbert space such that H C E continuously and densely. Identifying H with its dual H' by Riesz's isomorphism, we have

E'cHcE

(1.1)

where both embeddings are continuous and dense. In particular, it follows for the dualization E,(,) E : E' x E + R that

E,(l, h)E= ( I , h)Hfor all 1 E E', h E H.

177

Furthermore

H = { z E El sup{ ~ ' ( lz , ) 1 ~1 E E' with lllll.q ,< l} < co }. The norm in E and H we denote by Let

FCF

:= {g(ll,. . .

11 I I E

and

11

(1.3)

1J.qrespectively.

, l ~ 1 ) N E M,g E CF(WN),I I , . . . , l E~ E'},

(1.4) where C?(WN) denotes the set of all infinitely differentiable bounded functions with all derivatives bounded. For u E .FCr(RN)and z E E we define Vu(z)E H by all d (Vu(z), h)H= -&) := -u(z th)lt=o. (1.5) dt

+

Let p be a probability measure on ( E , B ( E ) )and denote the corresponding real LPspaces by U ( E ,p ) , p E [I,031, and define

For a set D of B(E)-measurable functions on E we denote the corresponding p-classes by

'6'. Throughout this paper we assume that the following hypothesis is fulfilled (Hl) If u E FC? such that u = 0 p a x . , then Vu = 0 p-a.e. and the (thus on L 2 ( E p , ) well-defined) positive definite symmetric bilinear form -P (&, 3 C r ) is closable on L 2 ( E ,p ) . Under condition (Hl) the Hilbert space H is sometimes called generalized CameronMartin space of p. We refer e.g. to [5] for the definition of closability and denote the -P closure of (€,,FC," ) on L 2 ( E , p )by (€P, H i 3 2 ( E , p ) ) Then . ( € P , H t g 2 ( E , p )is ) a symmetric Dirichlet form (see e.g. [5]).

Remark 1.1 (i) For sufficient conditions for (Hl) we refer to [2]. We note that those conditions are also necessary, if one requires all partial derivatives to be closable separately (see [2] for details). -P (ii) Closability of the form ( € P , F C) ~on L 2 ( E , p )is equivalent to the closability of the operator v : -PFC? c L 2 ( E , p )+L2(E+ H , p ) . We denote its closure (whose domain is, of course, H,"'(E, p ) ) again by V. If this operator is closable, then also for p 2 -P

V : FC? c L P ( E , p )+LP(E+ H , p ) is closable. Indeed, if un+ 0 in P ( E ,p ) as TI + 00 and (VU,),~N is a Cauchy sequence in P ( E + H , p ) , then the same holds in L 2 ( E , p )and L2(E + H , p ) respectively. By assumption it follows that Vu, + 0 in L2(E-+ H , p ) as n + 00, hence in p-measure, so by Fatou's Lemma

/11~u,ll~~dp < liminf m-m

s

I I V-~~u,~~gcip ,

But the right hand side can be made arbitrarily small.

178

(iii) Assuming that for p

V

>1

-P

:

F C p c L P ( E , p )+ LP(E4 H , p ) is closable,

(1.7) we can prove all what follows for p 1 instead of p = 2 with entirely similar proofs. For simplicity we restrict, however, to the case p = 2. The definition of capacities, however, we give for all p > 1 below. (iv) We refer t o [2] and [l]for examples for p satisfying (Hl). These examples include the white noise measure on E , i.e. the centered Gaussian measure on ( E , B ( E ) )with Cameron-Martin space H . But many other Gaussian measures and moreover Gibbs measures from statistical mechanics are included.

>

If for p E [1,co)condition (1.7) holds, we denote the closure by (V, H;’”(E,p)). For notational convenience we then set as usual for p 1

>

Now we recall the definition of capacity and intrinsic metric. Definition 1.1. (i) For U C E , U open, and p E [l,co),we set

Cl,,(U) := inf{lluIl;,p and for arbitrary A

IuEH

, ~ ~ ( E> , ~1 )p-a.e. , ~ on U )

cE

Cl,,,(A) := inf{CIJU) I A c V } . CI,“(A)is called capacity of A . ($a) A function f : A H R,A c E , is called C1,p-quasicontinuous if there exist closed sets A, C A, n E M, such that f I*,, is continuous for all n E M and limn+- Cl,”(E\A,) = 0.

Definition 1.2. For x , y E E set

A XY, ) := sup{f ( x ) - f ( Y ) I f p is called intrinsic metric of

(&p,

E FCr

with

llvf IIH < 1).

Hil,’(E,p ) ) .

The following is well-known. The proof is easy and included for the reader’s convenience. Lemma 1.3. Let x , y E E . Then

A x , Y) where we set l l z l l ~:= f c o if z E E \ H . Proof. Let f E FCP with I(Vf

= 115 - Y l l H ,

IIH < 1 and assume x - y E H . Then

Ilx-ylle. Here D f denotes the Frrkhet derivative of f . So,

179

So, by (1.3)

For A

c E as usual we set p ~ ( x:= ) inf{p(x,y) I y

E

A}, z E E.

Now we can formulate the main result of this section which we shall prove in the next section.

Theorem 1.4. Assume hypothesis ( H l ) holds. Let A E B ( E ) such that p ( A ) = 1. Then Cl,Z(PA> 0) = 0, i.e., the p-closure of A has full C17z-capacity. 2. PROOF OF THEOREM 1.4 Throughout this section hypothesis (Hl) is assumed to hold. Before we can prove Theorem 1.4, we need the following lemma.

Lemma 2.1. Let K C E be (1 . 11s-compact and c E (0,co).Then p~ is B(E)-measurable and p~ A c E H,”’(E,p) and I I V ( ~ A K c ) [ ~ H< 1.

Furthermore, p~ A c is C1,z-quasicontinuous.

Proof. Let {ei I i E N} C E’ be an orthonormal basis of H separating the points of E , and for n E M define Pn : E H En := span{el,. . . ,en} by n

P,Z :=

C E,(ei, z

) ei,~

z E E.

i=l

Fix y E E. By a simple approximation argument on EN we see that

u,(z)

:= IIP,z

Pnyll~A c , z E E ,

-

<

is a function in H;”(E,p) with I]Vu,ll~ 1. Clearly

un(x)T vv(x) := 11% - y l l A ~ c for all x

E E.

(2.1)

<

Hence by [5, Chap. I, Lemma 2.121 for y E E,vy E H t Y 2 ( E , p )with [Ivv,))~1 and un + v, weakly in H i Z 2 ( Ep, ) , hence the Cesaro mean of a subsequence converges strongly in H:’,’(E,p). A standard argument of Egorov type for capacities (cf. [5, Chap.111, Sect. 31) implies that selecting a subsequence if necessary, this Cesaro-mean converges C1,zquasiuniformly, i.e., uniformly on closed sets whose compliments have arbitrarily small J Cl,z-capacity. Hence by (2.1) vy is C ~ quasicontinuous.

180

Claim. Let dim E < 00. Then the assertions of the lemma hold even without assuming K to be compact. Since d i m E < 00, we have H = E and )I IIH and 1) JIE are equivalent norms. Let {yn I n E M} be a countable 11 . IlH-dense subset of K and defining vN:=inf{vy,,...vyN}, N E N , we have fK A

c = inf V N on E. N

Furthermore, (cf. e.g. [5, Chap. IV, Sect 41)

VN

E H ; ( E , p ) with

G

~ / V v h ’ ~ ~ HSUP{/IVVY1/lff7.” !I ~ V v y ~ ~ ~ H ) r

hence

( ( V V N (<~ H 1 for all N E N. Therefore, the claim follows by the same arguments as above. Now we go back to the general case. First we show that for all z E E PPnK(Pnx) T PK(x) n W. (So, in particular, p~ is B(E)-measurable.) Let z E E . Obviously, pp,K(Pnz) is increasing with n and S~PPP,K(J‘~X) < PK(Z) n

(cf. (2.1)). To prove the dual inequality we may assume that s u p n p p n ~ ( P n z<) 00. Let a E (0, co)such that supPP,K(PnZ) C: a. n

Then there exist kn E K such that

IIPnx - PnknllH < a for all n E M. (2.3) Since balls in H are weakly compact and K is compact in E , we can find a subsequence j-m j-+m k E K w.r.t. 11 )IE and Pn,z-Pnjknj h E H weakly in H . Hence such that knl for all i E M by (1.2)

-

-

Et(ei,h)E=

Jim E , ( e i , P n J x - P n j L j ) E

3-m

= lim E,(ei,z - knJ)E j-m

= E,(ei,z - k ) E .

Since {ei I i E M} separates the points of E , it follows that z - k = h and by (2.3) that llz - kJIH 6 liminf IIPnjz- Pn,knjIIH I--

< a.

(2.4)

In particular, pK(z) < 00. Now suppose that for some E E: (0, p ~ ( z ) ) supPp,K(Pnz) < PK (z)

- E E .

n

Then applying the above with a := p ~ ( z- )E , we get a contradiction from (2.4). So, (2.2) is proved. Now the assertions follow from the claim and the same arguments used for its proof. 0

181

Proof of Theorem 1.4. By inner regularity there exist compact sets K, such that p( K,) T p( A ) = 1. Let U,

:= PK, A 1,

c A, n

E M,

n E M.

Then u := inf un = lim u, 2

PA A

1 on E .

(2.5) Furthermore, u = 0 on Unal K,, hence u = 0 p-a.e. By Lemma 2.1 and the same arguments as in its proof we obtain that u is Cl,z-quasicontinuous, hence by [5, Chap. 111, Proposition 3.91 C1,2({u > 0)) = 0. But by (2.5), { P A > 0) C { u > 0) and the assertion is proved. 0 n

3. A RESULT

n-m

ON SETS WITH ZERO CAPACITY

Let { hl , hz, ' ' } be an ONB of H and let E, denote the linear span of { hl , . . . , h,} and set K := &En. A sufficient condition on A for p ( A K ) = 0 is given in [4]. Now we look for a condition which implies C,,l(A K) = 0. But we have to work under an additional quasi-invariance hypothesis. For k E H define

+

+

T ~ ( z ) : = z - ~z, E E .

(3.1)

(H2) For all k E H , p is k-quasi-invariant, i.e., p o r,kl we assume the Radon-Nikodym derivatives

p for all s E

R,and

to have the following properties: (H2a) afk 6 nq21Lq(E;p), for all s E R,and for all q E [l,co)the function s H llufkll, is locally bounded on R (H2b) For all compact C C R

ds < for p-a.e. z Lm denotes Lebesgue measure on P. 03

E E.

Here ds is Choosing appropriate versions by [l, Prop. 2.41 we may always assume that jointly measurable in s and z and that (H2b) holds for all z E E (rather than only p-a.e.

z E E). For examples of measures p satisfying condition (H2) we refer to 16, Section 31. As shown in [2] hypothesis (H2) implies (Hl). We need Sobolev spaces with differentiability index higher than 1. Analogue to the gradient operator V, we define the iterated gradient V2 on F C r by

Assume that -P

(H3) V2 : F C r C L P ( E , p )+ LP(E+ H 8 H , p ) is closable for p 2 1.

182

(3.3)

We define

and the fractional Sobolev spaces H:?' (1 < p < interpolation as follows.

03,

1 < r < 2) are defined by real

Definition 3.1. For 1 < p < co, 1 < r < 2 we define

H,"

=

(HA", Hi'p)r-l,p

(3.5)

where (., .) denotes the real interpolation space, see e.g. [3, 91. The norm in H,"P is given by the discrete K-method:

where

+

K(t,f) = inf{llfllll,p Ellf2/12,", fl + fz = f,f l E Hi,", fz E Hi,"} (3.7) It follows by a standard interpolation argument that H:" is uniformly convex (see e.g. [4])and we know from the denseness of FC," in HixPand [3, Th. 3.4.21 that FC? is dense in H:?'. A combination of these two facts implies that every u E H,'," has a C,,,-quasicontinuous redefinition which we denote by C. For 1 < p < 00, 1 r < 2 we define, for a [0, co]-valued lower semi-continuous functions h on E

<

C,,"(h) := inf{llull:,p; u E H:?',U

> h,a.e.},

and for an arbitrary [-co,co]-valued function f on E

C,,"(f) := inf{C,,P(h); h is 1.s.c.and h ( z ) 2 if(z)I Vz}. This definition is an extension of the previous one for sets in the sense that for any B

c E,

C,,p(B) = C r , p ( l B ) . The following result is parallel to Shigekawa [8] which is stated for Bessel capacities. We omit the proof which is the same as in 181. Note that (3.8) is implicit in IS].

-

Theorem 3.2. Fix q E (1,co),r E (1,2], P E (0,1] and 0 < y < P / q . Then there exists a constant C = C(q,r, P , y) such that If 5 : [O, TI" x ?!, R, ( t ,z ) H & ( z ) is measurable and if

then {&} has a version

{it}such that it is C,,,-quasi-continuous

Now we state the main result of this section

183

for every t E [0,TI" and

Theorem 3.3. Suppose (H2) and (H3) hold and let A C E , p > 1. If for any n there edsts a pair (rn,p n ) E (1,2]x ( 1 ,p ) with r, > np;' + 1 such that Cr,,p,(A) = 0, \Jn,

+

then Cl,p(A K ) = 0. Since capacities are continuous from below Theorem 3.3 immediately follows from Proposition 3.5 below. First we need a lemma. Fix n E M and define for t E [-M,MIn and f : E ++ R a function f ( t ) := f ( . Cy=ltihi).

+

Lemma 3.4. Fzx n E M and let p > q > 1 and r E (1,2]. Then there exzsts a constant C := C(p,q, r, T ) such that for all f E HOr,'

IIf(t)- f(s)lIl,q < CIlf IIr,pIt - 4 - l

(3.9)

for all s, t E [0,TIn. Proof. By the same argument as in [6], we can prove that for any T > 0, p > 1, q E (0,p) there exists a constant C1 = C,(p, q,T) such that for all f E and (s, t) E [-T,T]" x H;lp

[-T, TIn n

n

(3.10) Now let f E H;'

\ (0):

By (3.6),there exists a sequence ( f & ~

cH P

such that

(3.11) Choose the unique n E M such that

and the assertion is proved.

0

Proposition 3.5. Let T > 0, n E N,p > 1 and r E (1,2] such that r > np-' + 1. Let q E ( 1 , p ) such that r > nq-' + 1. Then there exists a constant C = C(n,p,q,r,T)> 0 such that for any A c W we have 3

G,q(A+ M ( T ;h l , . . . ,hn)) < C . C&(A) where

184

(3.13)

Proof. Set sh := c r = l s,h,. By changing signs we only need to prove

G,,(

u ( A+

sh))

< C . Cv?p(A).

sE[O,T]"

(3.14)

4

Let I denote the set of all rational points in [0,TI". If 0 c W is a% opei set then so is U s ~ [ O , T ] " ( O S h ) and We have

+

u

(0

+ sh) = U(0+ sh)

sE[O,T]"

SEI

Let eo denote the (r,p)-equilibrium potential of 0. Since eo 2 l o , we have

< We set f ( t ):= eo(.

Cl,,(suP eo(. + t h ) ) tEI

+ th). Applying Lemma 3.4 gives <

(3.15) Ilf(t)- f(s)lll,q Clleollr,plt- slr-l. For y E [0,T - 1 - nq-I), by Theorem 3.2 there exists a C1,,-quasicontinuous modification {&(.), t E [0, TI"} of { eo( . th),t E [0, TI"} such that

+

for some constant C = C(n,p,q,r,T)which may be different from that in (3.15). In particular, taking y = 0 we obtain Cl,,(SUP IEt - Ed sft

<

lleoll,4,p.

Hence

<

(C + 1) ' Ileoll:,p

= (C

+

1) ' C&(O). Thus (3.13) is proved for open sets. For general A, we have'for 0 3 A, 0 open

G,,(

u

(A+sh))

<

C1,,(

sE[O,T]"

u

(O+sh))

sE[O,T]"

+ 1) . C&(O)

(3.16)

+ 1) . inf{Crtp(0),o 3 A ) (C + 1) . Cr&(A),

(3.17)

<

(C

Consequently, c1,,(

U ( A+ sh)) <

sE[O,T]"

=

(C

as desired.

0

185

ACKNOWLEDGEMENT We would like to thank Terry Jegaraj for spotting a gap in the proof of Lemma 2.1 in the first version of this paper. Financial support of the BiBoS research center and the DFG through the Research Group “Spectral analysis, asymptotic distributions and stochastic dynamics”, as well as the DFG and the MPG through the project “SPDE’s with non-Gaussian white noise” is gratefully acknowledged.

REFERENCES [l]Albeverio, S., Kondratiev, Y. G. and Riickner, M. Ergodicity for the stochastic dynamics of quasiinvariant measures with applications to Gibbs states. J. Funct. Anal. 149 (1997), no. 2, 415-469. [2] Albeverio, S. and Rockner, M.: Classical Dirichlet forms on topological vector spaces--closability and a Cameron-Martin formula. J. Funct. Anal. 88 (1990), no. 2 , 395-436. [3] Bergh, J.; Lofstrom, J. Interpolation spaces. A n introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1576. [4]Hu, J. and Ren, J,: Infinite Dimensional Quasi Continuity, Path Continuity and Ray Continuity of Functions with Fractional Regularity. J. Math. Pures Appl. 80,1(2001),pp.131-152. 151 Ma, 2-M. and Rockner, M. Introduction to the theory of (nonsymmetric) Dirichlet forms. Universitext. Springer-Verlag, Berlin, 1992 [6] Ren, J. and Rockner, M.: Ray Holder-continuity for fractional Sobolev spaces in infinite dimensions and applications, Prob. Theory and Rel. Fields, 117 (ZOOO), n.2, 201-220. [7] Rockner, M. and Schied, A,: Rademacher’s theorem on configuration spaces and applications. J . Funct. Anal. 169 (1999), no. 2, 325-356. [8] Shigekawa, 1.: Sobolev spaces of Banach-valued functions associated with a Markov process, Probab. Th. Relat. Fields, 99 (1954), pp. 425-441. 191 Triebel, H.: Interpolation Theory, Function Spaces, Diflerential Operators, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1978.

186

AN INFINITE DIMENSIONAL LAPLACIAN IN WHITE NOISE THEORY DEDICATED TO PROFESSOR TAKEYUKI HIDA ON THE OCCASION OF HIS 77TH BIRTHDAY KIMIAKI SAITO Department of Mathematics Meijo University Nagoya 468-8502, Japan E-mail: ksaitoOccmfs.meij0-u.ac..jp The Ldvy Laplacian is formulated as an operator acting on a class in the white noise L2 space consisting of functionals of Gaussian noise and Poisson noise. This class includes regular functionals in terms of Gaussian noise and it is large enough to discuss the stochastic process generated by the Ldvy Laplacian. This formulation is slightly outside of the usual white noise distribution theory, while the Levy Laplacian has been discussed within the framework of white noise analysis. From Cauchy processes an infinite dimensional stochastic process is constructed, of which the generator is the Ldvy Laplacian.

Mathematics Subject CIassifications (2000): 60H40 Introduction In 1922 P. L6vy l 7 introduced an infinite dimensional Laplacian on the space L2(0,1).Since then this exotic Laplacian has been studied by many authors from various aspects see [l-6,18,20,23] and references cited therein. In this paper, generalizing the methods developed in the former works [16,19,24,28,29], we construct a new domain of the Lkvy Laplacian acting on some class of functionals of Gaussian noise and Poisson noise, and associated infinite dimensional stochastic processes. This paper is organized as follows. In Section 1 we summarize basic elements of white noise theory based on stochastic processes given as linear combinations of Brownian motions and Poisson Processes. In Section 2, following the recent works Kuo-Obata-Sait6 16, Obata-Sait8 24,Sait6 30 and Sait8-Tsoi 31, we formulate the L6vy Laplacian acting on a space consisting of white noise functionals of linear combinations of Gaussian noise and Poisson noise, and give an equi-continuous semigroup of class (CO)generated by the Laplacian. This situation is further generalized in Section 3 by means of a direct integral of Hilbert spaces. The space is enough large to discuss the stochastic process generated by the Ldvy Laplacian. It also includes regular 187

functionals (in the Gaussian sense) as a harmonic functions in terms of the LCvy Laplacian. In Section 4,based on infinitely many Cauchy processes, we give an infinite dimensional stochastic process generated by the Lkvy Laplacian. 1

White noise functionals of Gaussian noise and Poisson noise

Let E = S(R)be the Schwartz space of rapidly decreasing R-valued functions on R. There exists an orthonormal basis {e,},~o of L2(R) contained in E such that d2 + u 2 1. du2 For p E R define a norm 1 * , 1 by = ( A P f l p ( R )for f E E and let Ep be the completion of E with respect to the norm I ., 1 Then Ep becomes a real separable Hilbert space with the norm 1 ., 1 and the dual space E; is identified with E-, by extending the inner product of L2(R) to a bilinear form on E-, x E,. It is known that

Ae,

= 2(v+

l)e,,

. .,

A = --

Y = 0,1,2,.

+

If/,

(a,

a)

E* = indlimE-,.

E = projlimp,,Ep,

,-+a,

The canonical bilinear form on E* x E is also denoted by (., .). We denote the complexificationsof L2(R), E and Ep by L&(R),Ec and Ec,,, respectively. Let {B(t)}t>obe a Brownian motion and {N(t)}t>oa standard Poisson process which is-independent of the Brownian motion, For any u 2 0 and X E R, we set A,,x(t) = uB(t) XN(t), t 2 0. Then we have

+

n

~ [ e ' " * - ~ ~= ( t eth(z), )] h ( z ) = --z2 uL 2

+ eiX%

- 1,

t 2 0.

Set C([)= exp {JR h ( < ( u ) ) d u }, [ E E . Then by the Bochner-Minlos Theorem, there exists a probability measure po,x on E* such that

s,*

exp{i(z,E)} dcLu,x(z) = W,

E

E E.

Let (L2>,,x= L 2 ( E * p, , , ~ )be the Hilbert space of C-valued square-integrable functions on E* with L2-norm 11 . I,x with respect to p , , ~ .The Wiener-It6 decomposition theorem says that:

n=O

188

where Hn is the space of multiple Wiener integrals of order n E N and HO = C. According to (1.1)each 'p E (L2),,,x is represented as

n=O

where L&(R)63" denotes the n-fold symmetric tensor power of Lg(R) (in the sense of a Hilbert space). An element of ( L 2 ) , , ,is~ called a white noise functional. We denote by .)),,A the inner product on ( L 2 ) , , ~Then, . for 'p and $J E (L2),,,0 we have ((e7

x L&(R)""is denoted also by where the canonical bilinear form on L&(R)@'" .). For cp and $J E (L2)0,xwe also have (.7

00

00

n=O

n=O

00

n=O

The U-transform of cp E (L2),,,xis defined by

Theorem 1.1 26 (see also Let F be a complex-ualued function defined o n E . Then F is a U-transform of some white noise functional in (L2),,,~if and only if there exists a complex-valued function G defined o n E c such that 9914922)

1 ) f o r any z E

c,

< and q in E c , the function G(zJ+ q ) is an entire function of

2) there exist nonnegative constants

K and a such that

IG(t)I I Kexp [alrl:] 7

+

3) F ( [ ) = G(io2( X(eiAt - 1))f o r all

189

t E E.

vt E Ec,

2

The LQvy Laplacian acting on white noise functionals

Consider F = U p with 'p E (L2),,x. By Theorem 1.1, for any t,rl E E x E the function z H F ( [ z q ) admits a Taylor series expansion:

+

where F ( " ) ( [ ): E x . . x E + C is a continuous n-linear functional. Fixing a finite interval T of R, we take an orthonormal basis c E for L 2 ( T ) which is equally dense and uniformly bounded (see e.g. Let V Ldenote the set of all 'p E (L2),,x such that the limit '*I'~).

exists for any E E E x E and i ~ ( U ' pis) in U[(L2),,x].The LQvy Laplacian A, is defined by

ALP = U - l Z i ~ U ' p ,

'p

E V)t.

Given (I 2 0,X E R, n E N and f E L & ( R ) 6 n ,we consider of the form:

'p

E

(L2)u,x

r

The U-transform U'p of

'p

is given by n

where E o , x ( [ ) ( u j )= ia2[(uj)+XeiXt(uj). For any (I 2 0, X E R and n E N let E,,x,, denote the space of 'p which admits an expression as in (2.1), where f belongs to E t " and supp f c T". Then, since the space E , , A ,is~included in (L2),,x, we define E,,x,~ by the completion of E u , ~ ,with n respect to 11 . II,,x. Set E,,x,O = C for any (I 2 0, X E R. Then E , , A ,is~a closed linear subspace of (L2),,x. Using a similar method as in 31, we get the following

Theorem 2.1 31 (see also For each (I 2 0 , n E N and X E R the LLvy Laplacian AL becomes a scalar operator on E,,o,,,U Eo,~,, such that l6l2')

AL'p =

0,

'p €

190

E'o,o,n;

(2.2)

Proposition 2.2 of the form:

31

Let X E R be fixed. Consider two white noise functions

M

M

n=O

n=O

If 'p = .IC, in (L2)0,x,then p n

= &, for all n E

N U (0).

Taking (2.2) and (2.3) into account, we put

For N E N and X E R let D Y be the space of

'p E

(L2)o,xwhich admits an

expression 00

'p

=

C

p n E Eo,x,n,

qn,

n=l

such that 00

n=l

By the Schwartz inequality we see that D Y is a subspace of (L2)0,x and becomes a Hilbert space equipped with the new norm 111. IIIN, o , x defined in (2.4). Moreover, in view of the inclusion relations:

( L 2 ) o , x3 D:)x

3

.. . 3 D Y

2 D$tl 3

... ,

we define

nD y . 00

D%x= projlimN,OO DOJ=

N=l

Note that for any X E R we have 00

IJ Eo,x,n c ~2~c ( ~ ~ ) o , x . n=l

191

OX

Then AL becomes a continuous linear operator defined on D i + l into D$x satisfying

IllALcplllN,O,A5

IJIcpIIIN+l,O,X,

cp E

DLA, N

E N.

(2.5)

Summing up, we have the following

Theorem 2.3 l 6 t 3 l The operator AL is a self-adjoint operator densely defined in D$x for each N E N and X E R.

It follows from (2.5) that AL is a continuous linear operator on D%x.In view of the action of (2.3), for each t 2 0 and X E R we consider an operator G; on DLx defined by 00

00

n=l

n=l

We also define G: on ( L 2 ) g as , ~ an identity operator I by Icp = cp, cp E

(L2 )u,o. Theorem 2.4 16,30 Let X E R. Then the family of operators { G ; ; t 2 0 ) o n DZAis an equi-continuous semigroup of class ((70)of which the infinitesimal generator is A,. 3

The LQvy Laplacian acting on WNF-valued functions

Let dv(X) be a finite Bore1 measure on R satisfying

Fix N E N. Let DO, be the space of (equivalent classes of) measurable vector functions cp = (pX)with cpx = C,"=lcp; E D Y for all X E R \ {0}, and cpo E ( L 2 ) , , 0 , such that

Then !DO, becomes a Hilbert space with the norm given in (3.1). Take a functional

192

j = ( j l , . . . ,jn)E (N u

{[n}r=o

{O}Y,

where an orthonormal basis c E for L2(T)which is equally dense and uniformly bounded. Since a linear span of { ' p t f ) ; nE N u {O},j = (jl, . . . ,jn) E (N U { O } ) n } is dense in D Y and a linear span of {cpzf';n E N U { 0 } , j = (jl,. . . ,jn) E (N U {O})n} is dense in (L2),,,0,the spaces 9%is nothing but a direct integral Hilbert space:

+ C,"==, sR\{o} q: dv(X)

Let 9: be the space of white noise functionals 'po with C,"=l p ' : E (L2)0,xfor all X E R \ {0}, and 'po E

( J ~ ' ) ~ , O such ,

that

Then 9; also becomes a Hilbert space with the norm given in (3.2). Proposition 3.1 The map

is a continuous linear map and a bijection from 9%into 9;. In view of the natural inclusion: 9%+1 c 9%for N E N, which is obvious from construction, we define

n 00

9.0,= projlimN+009G =

9%.

N=l

The LQvyLaplacian AL is defined on the space 9& by

ALcp = (AL'pX), cp = ((PA) E 9 .: Then A, is a continuous linear operator from D.", into itseIf. Similarly, for t 1 0 we define Gtcp = ( G W ) , Then we have the following:

cp = ((PA) E 9.0,.

Theorem 3.2 The family { G t ; t 2 0) is a n equi-continuous semigroup of class (CO)o n D.", whose generator is given by AL.

193

4

An infinite dimensional stochastic process associated with the LBvy Laplacian

For p E R let EF be the linear space of all functions X H Ex E E,, X E R, which are strongly measurable. An element of EF is denoted by E = ( E A ) A ~ R . Equipped with the metric given by

the space EF becomes a complete metric space. Similarly, let CR denote the linear space of all measurable function X H zx E C equipped with the metric defined by

Then CR is also a complete metric space. In view of d, 5 d, for p 2 q, we introduce the projective limit space ER = projlim,,, EF. The U-transform can be extended to a continuous linear operator on D& by

E = (EX)XER E ER, UP(() = (U(pX(&+))XR, for any cp = ((P’)X€R E 92. The space U[!D&]is endowed with the topology induced from D& by the U-transform. Then the U-transform becomes a homeomorphism from D& onto U[D&]. The transform Ucp of cp E 9: is a continuous operator from ER into CR. We denote the operator by the same notation U p . Let

Et

be an operator defined on U[DZ] by Et = UGtU-I,

t 2 0.

Then by Theorem 3.3, {Ct;t 2 0) is an equi-continuous semigroup of class (Co)generated by the operator

LL.

Let {Xi}, j = 1 , 2 , be independent Cauchy processes with t running over [0, oo),of which the characteristic functions are given by

E[ei”xi]= e-tlzl , Take a smooth function VT E E with =

ZER, j=1,2. VT = 1/ITI

on T . Set

XitqT if X 2 0, X ? , , T ~ , otherwise.

194

Define an infinite dimensional stochastic process {Yt;t 2 0 } starting at ( 6 X ) X E R E E R by

Yt = ( E X

+q X ) X E R ,

<=

t 2 0.

Then this is an ER-valued stochastic process and we have the following

Theorem 4.1

If F

-

is the U-transform of an element in DZ, we have

GtF(S)= E [ F ( Y t ) l Y o=
t 2 0.

PROOF.We first consider the case when F

E U[D&] is

(4.1)

given by

F(<) = (FX(&))XER,Fo E u[(L2)c,0],

with f E E$". Then we have

= (G;FX( h ) ) X E R -

XER

= GtF(<).

Next let F = (FoGx,o-tCz=lF,")XERE U[9:]. Then for v-almost all X E R and for any n E N, F," is expressed in the following form: n

F,X(JX)

= lim A" N-CC

Since Fo E U [ ( L 2 ) c ,and ~ ] F," E U[DLX],there exist 'po E ( L 2 ) o ,and ~ E DZXsuch that Fo = U [ p 0 ]and F," = U['pA]for v-almost all X and each n. By the Schwarz inequality, we see that

195

n=l W

-

where 'pcx = C(&,)-lei('*~X) for v-almost all X E R and each N E N. Therefore by the continuity of G;, X E R, we get that

/ W

\

-

= GtF(5).

Thus we obtain the assertion.

I

Acknowledgments This work was written based on a talk at International Conference on Stochastic Analysis: Classical and Quantum - Perspectives of White Noise Theory. This work was partially supported by JSPS grant 15540141. The author is grateful for the support.

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196

2. L. Accardi, P. Gibilisco and I. V. Volovich: Yang-Mills gauge fields as harmonic functions for the Lkvy Laplacian, Russ. J. Math. Phys. 2 (1994), 235-250. 3. L. Accardi and 0. G. Smolyanov: Trace formulae for Levy-Gaussian measures and their application, in “Mathematical Approach to Fluctuations Vol. I1 (T. Hida, Ed.),” pp. 31-47, World Scientific, 1995. 4. D. M. Chung, U. C. Ji and K. Sait6: Cauchy problems associated with the Le‘vy Laplacian in white noise analysis, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 2 (1999), 131-153. 5. M. N. Feller: Infinite-dimensional elliptic equations and operators of Lkuy type, Russ. Math. Surveys 41 (1986), 119-170. 6. K. Hasegawa: Le‘vy’s functional analysis in terms of an infinite dimensional Brownian motion I, Osaka J. Math. 19 (1982), 405-428. 7. T . Hida: “Analysis of Brownian Functionals,” Carleton Math. Lect. Notes, No. 13, Carleton University, Ottawa, 1975. 8. T. Hida: A role of the L h y Laplacian in the causal calculus of generalized white noise functionals, in “Stochastic Processes,” pp. 131-139, SpringerVerlag, 1993. 9. T. Hida, H.-H. Kuo, J . Potthoff and L. Streit: “White Noise: An Infinite Dimensional Calculus,” Kluwer Academic, 1993. 10. T. Hida and K. Sait6: White noise analysis and the LLvy Laplacian, in “Stochastic Processes in Physics and Engineering (S. Albeverio et al. Eds.),” pp. 177-184, 1988. 11. E. Hille and R. S. Phillips: “Functional Analysis and Semi-Groups,” AMS Colloq. Publ. Vol. 31, Amer. Math. SOC,1957. 12. I. Kubo and S. Takenaka: Calculus on Gaussian white noise I-IV, Proc. Japan Acad. 56A (1980) 376-380; 56A (1980) 411-416; 57A (1981) 433-436; 58A (1982) 186-189. 13. H.-H. Kuo: O n Laplacian operators of generalized Brownian functionals, Lect. Notes in Math. Vol. 1203, pp. 119-128, Springer-Verlag, 1986. 14. H.-H. Kuo: “White Noise Distribution Theory,” CRC Press, 1996. 15. H.-H. Kuo, N. Obata and K. Sait6: Lkuvy Laplacian of generalized functions o n a nuclear space, J. Funct. Anal. 94 (1990), 74-92. 16. H.-H. Kuo, N. Obata and K. Sait6: Diagonalization of the Lkuy Laplacian and related stable processes, Infin. Dimen. Anal. Quantum Probab. Rel. TOP.5 (2002), 317-331. 17. P. LQvy: “Lecons d’Analyse Fonctionnelle,” Gauthier-Villars, Paris, 1922. 18. R. Lkandre and I. A. Volovich: The stochastic Lkvy Laplacian and YangMills equation o n manifolds, Infin. Dimen. Anal. Quantum Probab. Rel. TOP.4 (2001) 161-172.

197

19. K. Nishi and K. Sait6: An infinite dimensional stochastic process and the Lkvy Laplacian acting on WND-valued functions, to appear in “Quantum Inofrmation and Complexity” World Scientific, 2004. 20. K. Nishi, K. Sait6 and A. H. Tsoi: A stochastic expression of a semigroup generated by the Lkvy Laplacian, in “Quantum Information I11 (T. Hida and K. Sait6, Eds.),” pp. 105-117, World Scientific, 2000. 21. N. Obata: A characterization of the Lkvy Laplacian in terms of infinite dimensional rotation groups, Nagoya Math. J. 118 (1990), 111-132. 22. N. Obata: “White Noise Calculus and Fock Space,” Lect. Notes in Math. Vol. 1577, Springer-Verlag, 1994. 23. N. Obata: Quadratic quantum white noises and Lkvy Laplacian, Nonlinear Analysis 47 (2001), 2437-2448. 24. N. Obata and K. Sait6: Cauchy processes and the Lkvy Laplacian, Quantum Probability and White Noise Analysis 16 (2002), 360-373. 25. E. M. Polishchuk: “Continual Means and Boundary Value Problems in Function Spaces,” Birkhauser, Basel/Boston/Berlin, 1988. 26. J. Potthoff and L. Streit: A characterization of Hida distributions, J. Funct. Anal. 101 (1991), 212-229. 27. K. Sait6: Its’s formula and Lkvy’s Laplacian I, Nagoya Math. J. 108 (1987), 67-76; 11,ibid. 123 (1991), 153-169. 28. K. Sait6: A (Co)-group generated by the Ldvy Laplacian II, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 1 (1998) 425-437. 29. K. Sait6: A stochastic process generated by the Lkvy Laplacian, Acta Appl. Math. 63 (2000), 363-373. 30. K. Sait6: The Lkvy Laplacian and stable processes, Chaos, Solitons and Fractals 12 (2001), 2865-2872. 31. K. Saitb and A. H. Tsoi: The Lkvy Laplacian as a self-adjoint operator, in “Quantum Information (T. Hida and K. Sait6, Eds.),” pp. 159-171, World Scientific, 1999. 32. K. Sait6 and A. H. Tsoi: The Lkvy Laplacian acting o n Poisson noise functionals, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 2 (1999), 503-510. 33. K. Sait6 and A. H. Tsoi: Stochastic processes generated by functions of the Lkvy Laplacian, in “Quantum Information I1 (T. Hida and K. Sait6, Eds.),” pp. 183-194, World Scientific, 2000. 34. K. Yosida: “Functional Analysis (3rd Edition) ,” Springer-Verlag, 1971.

198

INVARIANCE OF POISSON NOISE

SI SI Graduate School of Information Science and Technology Aichi Prefectural University Nagakute, Aichi-ken 480-1198, Japan

ALLANUS TSOI Department of Mathematics University of Missouri Columbia, MO 65211, U.S.A. WIN WIN HTAY Department of Computational Mathematics University of Computer Studies Yangon, Myanmar

Dedicated to Professor Talceyulci Hida on the occasion of his 77th birthday

AMS Mathematics Subject Classification 2000 : 60H40

We discuss some transformation groups under which Poisson noise is kept invariant, so that we can give a characterization. The idea is similar to the case of Gaussian white noise, however the actual transformation group is quite different. In fact, symmetric group plays important role in the case of a Poisson noise.

1. Introduction Our main purpose is to find out a characterization of Poison noise from the viewpoint of white noise analysis. For this purpose we refer to the characterization of white noise, using the infinite dimensional rotation group O ( E ) . See e.g. [5]. We are therefore suggested to find a suitable transformation group acting on the sample function space of a Poisson noise. 199

Symmetric groups of all orders are fitting for our purpose. In addition, some other groups that give invariance of Poisson noise tell us finer properties of Poissson noise and Poisson process. 2. Preliminaries

Introduce a Poisson noise by taking the parameter space to be Rd. A construction is as follows. Let El and E2 be suitably chosen Hilbert spaces that are topologized by Hilbertian norms 11 111 and (1 . 112, respectively. Assume that 11 5 11 . 112 and that there exist consistent injections Ti,i= 1 , 2 , such that Ti from Ei to Ei-1 for i = 1 , 2 , is of Hilbert-Schmidt type, where EO = L 2 ( R d ) Then, . we have inclusions of the form

E2 C El

c Eo c E; c E l .

(2.1)

Recall that the characteristic functional of Poisson noise with Rdparameter is given by

where we take unit intensity. It can be proved that C$ ( I )is continuous on El . It is positive definite and Cg(0) = 1. The Bochner Minlos theorem guarantees the existence of a probability measure p p on E i such that its characteristic functional is equal to the given C$(<).Thus a Poisson noise measure space (E;,p p ) is given. To fix the idea, let the time parameter space be a compact set, say is continuous in EO c L 2 ( I ) ,so that there is a Gklfand triple of the form

I

= [0,1].In this case, C;(<)

El

c L2(1)

Eo

c E;.

(2.2)

A Poisson measure is now introduced on the space E;. Define P ( t , z ) = (z,xpt]),O5 t 5 1 , z E Ei,by a stochastic bilinear form. Then, P ( t ,x) is a Poisson process with parameter set [0,1]. Let A, be the event on which there are n jump points over the time interval I. That is

A, = {z E E ; ; P ( l , z )= n}, 200

(2.3)

where n is any non-negative integer. Then, the collection {A,, n 2 0) is a partition of the entire space E;. Namely, up to measure 0, the following relations hold:

AnnA,=d, n#m; UAn=E;.

(2.4)

The conditional probability on A, is

For C c Ak, the probability measure is defied as

on a probability measure space (Ak,Bk, &), where Bk is the sigma field generated by measurable subsets of Ak, determined by P ( t ,x).

For k = 0, the measure space is (Ao,Bo, &) is trivial, where

Bo

= {d,Ao} mod

&

and &(A) = 1. We now recall a general notions about a probability measure space.

Definition A probability space (0,B , P ) is a Lebesgue space if 1) I = {I,} is a countable base such that

B = o{I,;n E 2). 2) Let A, = I, or 1:. Then n,A, # 4. 3 ) For 5 # y, there exists I,, I , such that I,nI,

=4

, 5 E I,,

y E I,.

It is known that if (R, B , P ) be a Lebesgue space then it is isomorphic to ([O, a ] ,Leb) u {atoms}.

Lemma 1. On a Lebguse space a measure preserving set transformation implies a measure preserving point transformation. Coming back to a Poisson noise with parameter set [0,1].

Proposition 1. (Ak,Bk, &), k 2 1 is a Lebesgue space without atom. 201

= ~i(x),i=

Let x E An,n 2 1, and let 7i statistics of jump points of P ( t ) :

1 , 2,..., n, be the order

0 = 70 < 7-1 < . . . < 7, < 7,+1 = 1. (The 7's are strictly increasing almost surely.) Set

xi(.)

= 7i(X) - Ti-i(X),

so that

cxi

n+l

= 1.

1

Proposition 2. On the space A,, the (conditional) probability distribution of the random vector (X1,Xz, ...,X,+l) is uniform on the simplex n+l

j=l

Cororally The probability distribution function of each Xj is 1 - (1 - U ) n ,

0 Iu

I 1.

Proposition 3. The conditional characteristic functional

Cp,,(E) = E [ e i ( P IIAn] E) is obtained as

3. Invariance of p p under transformations on parameter space We are going to find a transformation group which is acting on the Poisson noise space ( E * ,,up) and which keeps the ,up invariant. Note that in Gaussian case the actions are taken on sample functions B(t),such as

x E E*,of

g* : x

4

202

g*x.

For the case of Poisson noise, in contrast with the Gaussian case, we begin with measure preseving set transform, then we come to transformations of sample functions. Also, it is noted that our probabilistic approach is also different. With the study of invariance of Poisson noise measure we aim at harmonic analysis of functionals of Poisson noise, where the invariance of Poisson noise will play interesting roles. In particular, the family of symmetric groups acting on the conditional probability space is a characteristic of Poisson noise. There are two ways to introduce such transformation group. One uses the vector space structure of E* with the help of a base of E*. Another way comes from the transformation acting on the parameter space. This means that the second method, unlike the first method, depends on the geometric structure of the space in question.

3.1. Transformations in the Rd-parameter case We give illustrative examples in the followings.

Class 1. Use of an orthogonal system Let {Vn} be a system of unit cubes such that UV, = Rd and that Vn's are disjoint except boundaries.

~ ( t=)C E n ( t ) , a.e.7

E

E

(3.1)

n

where En(t) = E ( ~ ) x v , ( ~ ) . Let 7r be permutation of finitely many n. and gn be the transformation such that 00

(g7rE)(t) =

CElr(n)(t) E L2(R").

(3.2)

-m

Then we can see that the characteristic functional of Poisson noise satisfies

Hence the two characteristic functionals define the same Poisson measure. 203

Case 2. Group of motions We can easily see that the time shift leaves the probability distribution p p of a Poisson noise invariant, since

cp(s,"<) = c p ( < ) , (st<)(u) = <(u-tek),

E Rd,

(3.3)

where ek is the unit vector in the k-th direction of Rd. Obviously, orthogonal group O ( d ) acting on the parameter space Rd presents invariance of Poisson measure. Thus we have

Proposition 4. The probability distribution p p of an Rd-parameter Poisson noise is invariant under 1) the rotation group S O ( d ) acting on Rd, and 2) the shifts. Note. Interesting property can be seen in the dilation of Rd-vectors: rt : u -.+ ueat, u E Rd, a > 0.

(3.4)

Then the characteristic functional Cp(c) changes to

Namely, the intensity changes from X to Xe-dat, but remains to be the Poisson distribution.

3.2. Td-parameter case Automorphisms of T d Take the parameter space to be a compact domain, say T d , d dimensional Torus instead of Rd in (2.1). Then we have a corresponding GBlfand triple

El

c L 2 ( T d ) = Eo c E;.

(3.6)

The characteristic functional of Poisson noise with T d parameter is

204

In particular, if we take d = 1then T d is S1, a circle and the characteristic functional is

where the intensity X is taken to be 1. Let r be a transformation defined by (r)(t) = 2t(mod 2n),t E S1, and define the automorphism a of E by (a<)(u) = t(7(u))* We can see that CP,1(05)= CP,1(5).

It means that the characteristic functional is a-invariant and then measure p p is a*-invariant.

Proposition 5. The mapping 0 is an automorphism of uous. There esists a* acting on EF such that

El

and is contin-

D*PP = PP and un*pp = p p .

4. Transformations that preserve conditional Poisson measures

+

Let Sn+l = {7rn(i),1 5 i 5 (n l)!}be the symmetric group and B be the set of Bore1 subsets of simplex An+, where n+l

An+, = {(x11x21*.*,xn+ll

Cxi = 110 I xi 5 1). 1

Let

X=

(X1,X21...lxn)

205

and set

X7rn(i)= (xTy(i),XT;(i), .*‘,xT_n+l(i)), where

7rn(i)

= ( ~ r ( i ) , 7 r g ( i ) , .,~:+~(i)) ..

E &+I.

Let g7rn(i) be the set

transformation on E* such that gTn(i) :

x-’(B) X$(il(B),B -+

E

B(&+i).

(X-l(B),X;:(i)(B) c A,.) We can see that

P;(X-l(B)) = P;(x;:(i)(B)), and g T n ( i ) defines an automorphism of (A,, p;). Then an automorphism g T ( i ) on E* is defined by the automorphisms g T n ( i ) on A,, n = 1 , 2 , .. . , such that gT(i)a:= g , p ( i ) Z ,

z E A,,

n = 0 , 1 , 2 , ....

The space ( A k , Bk, p $ ) is a Lebesgue space by Proposition 1. Thus by Lemma 1, gTk(i) defines a measure preserving point transformation on ( A k , Bk,p $ ) . The same for g r ( i ) . Then we have

Proposition 6. The transformation g T ( i )is a pp-measure preserving point transformation on E*.

Theorem 1. The group G defines a symmetry of Poisson noise. Applications to unitary representation can be given to develop a harmonic analysis, and a comparison with the Gaussian case will be discussed in the forthcoming paper by the author.

Remark We can play the same game if the parameter set is [ O , l l d . 206

5. A characterization of Poisson noise Start with a probability space (G,,P,) and let {Y,(t,w),w E G,} be a stochastic process on (G,, P,), such that n

Yn(t,~)=C& t E~ [ O(, l~ ] , )O(=tt ) o <, t l < . . . < t , = l .

(5.1)

1

Theorem 2. Assume that (1) the joint probability distribution of n singular points t j ( w ) , j = 1,..n,are absolutely continuous, (2) {Y,(t, w ) } and {Y,(T(~), w ) } have the same probability distribution, where T is the transformation defined as

~ ( t=)2t (mod 1).

(5.2)

Then the probability distribution of singular points of Yu(t,w ) on the space G, is the uniform distribution. (Note that in Section 3.2, T is defined by mod 27r. Now we use mod 1.) Proof. Let p(t1, t 2 , . . ,t d ) be the joint probability distribution of singular points t l , t 2 , ,t d . Use the transformation T , for all of t l , . . . ,t d , for which p(t1,. . . , t d ) is invariant, then we have +

After N iteration, letting N tend to infinity we have P ( t l , . . . , t d ) = i l i l P ( U l ,... ,Ud)dUl’..dW,

Thus the assertion is valid. The proof is the same for any n > 1.

Proposition 7. The characteristic functional C,(E) is invariance under the transformation G, where CJ is defined by ( a ( ) ( u )= <((.‘(u)) in which 7‘ be a transformation defined by l ( t )= (T(tl), . . . ,~ ( t d ) for ) t = (tl... . ,t d ) E [0, lIdsuch that ~ ( t j= ) 2tj(mod

207

l), t j E [0,1].

Theorem 3. The characteristic functional of Yn(u),on the probability space G, with the parameter space (time interval) [0,1], is (5.4)

We now consider the parameter space [0,TI instead of [0,1],at the aim of extending the parameter space to [ O , o o ) .

Corollary The characteristic functional of Yn(t),on the probability space G, with the parameter space [0, TI, is expressed in the form

Set R = EnG,. Let us define a stochastic process Y ( t )on (0,P ) such that

is additive and Levy process. In order to have Y ( t )well defined, it is necessary to determine the probabdity of G,. Let P(Gn) = en. The characteristic functional functional C(<;O,s)of Y ( t )on R, with the parameter space [0, s], is

since Cn([;0, S ) = E[eic(tJGn]. From the above assumption (ii), Y ( t )is additive then we have

C(<;0, s1

+ s2) = C(<;0, s1)C(I;

The relation (5.5), (5.6) and (5.7) yields

208

S1,Sl

+ s2).

(5.7)

Consequently, we have

It is written as

where

Equating the coefficient of a1bn-', we have

n!dn(sl

+ SZ) = ( l ! d l ( s l ) )( ( ~ - L ) ! & - ~ ( S Z ) )

By letting h k = k ! d k ( t ) ,we obtain that ho = 1,hk where X is a constant, Hence, we are given

*

=

Xk,k

=

1 , 2 , . . .,

Since C k = e-xt, we have to normalize them to be a probability distribution. Thus we have

That is, Y ( t )has a Poisson distribution. Theorem 4. Under the above assumptions (i) and (ii), Y ( t )is a Poisson noise.

Proof. From (5.6) and (5.8) the characteristic functional of Y ( t )is

209

References 1, L. Accardi, T. Hida and Si Si, Innovations for some stochastic processes. Volterra Center Notes, 2002. 2, T. Hida and Si Si, Innovation approach to some random fields, Application of white noise theory, World Scientific (2004). 3, T. Hida , Canonical representation of Gaussian processes and their applications, Memoirs Coll. Sc., Kyoto A33 (1960) 109-155 4. T. Hida, Stationary Stochastic Processes, Math. Notes, Princeton Univ. Press. 1970. 5. T. Hida, Brownian motion, Springer-Verlag, 1980. 6. P. Lkvy, Fonctions aleatoires A corrklation linebire, Illinois J. of Math. 1 (1957), 217-258. 7. S. Mataramvura, B. Oksendal and F. Proske, The Donsker delta fuction of a Lkvy process with applicatio to chaos expansion of local time, Annales de Institut H. Poincark, Elsevere SAS, 40 (2004) 553-567. 8. V. A. Rohlin, On the fundamental ideas of measure theory, AMS Transaction series 1. vol. 10 (1962), 1-54. 9. Si Si, Effective Determination of Poisson noise, Infinite Dimensional Analysis and Quantum Information vo1.6. Number 4, World Scientific Pub. Co. (2003), 609-617. 10. Si Si, Note on Poisson noise, Quantum Information and Complexity, World Scientific 2004, 411-425. 11. Si Si and Win Win Htay, Structure of Linear processes, Proceeding of International conference on Quantum Information, (to appear). 12. Win Win Htay, Note on Linear process, Quantum Information and Complexity, World Scientific 2004, 449-455.

210

NONEQUILIBRIUM STEADY STATES WITH BOSE-EINSTEIN CONDENSATES S. TASAKI* AND T. MATSUIt *Advanced Institute for Complex Systems and Department of Applied Physics, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan Graduate School of Mathematics, Kyushu University, 1-10-6 Hakozaki, Fukuoka 812-8581, Japan Nonequilibrium steady states (NESS) of bosonic system with Bose-Einstein condensate are investigated with the aid of the C*-algebraic method. The system consists of two free bosonic reservoirs coupled with each other. Initially the reservoirs are prepared to be in equilibrium with different temperatures and local densities. NESS are constructed as the t + +alimits of such initial states. Josephson currents are studied as well.

1 Introduction The understanding of irreversible phenomena including nonequilibrium steady states is a longstanding problem of statistical mechanics. Various theories have been developed so far and one of promising approaches deals with infinitely extended dynamical systems1,2. Not only equilibrium properties, but also nonequilibrium properties has been rigorously investigated. Those include analytical studies of nonequilibrium steady states of harmonic crystal^^)^, a one-dimensional gas5, unharmonic chains6, an isotropic XY-chain7, a onedimensional quantum conductor', systems with asymptotic a b e l i a n n e s an interacting fermion-spin systemll, fermionic junction systems12, a quasispin model of superconductors13 and a bosonic junction system without the Bose-Einstein condensate14. Entropy production has been rigorously studied as well (see [9,15-191 and the references therein). In this article, we add more examples of nonequilibrium steady states (NESS), namely bosonic systems with Bose-Einstein condensate. The paper is arranged as follows: In the next section, the a bosonic junction system is described. In Sec. 3, its NESS are constructed and their properties are studied. In Secs. 4 and 5, NESS for free Bosons on lattice with different joints are investigated. The Josephson currents at such NESS are studied as well.

2

Bosonic Junction System

In this and the next sections, we consider a bosonic version of a typical junction system studied in [la]. The model is defined on a tensor product of a 211

Hilbert space L 2 ( R ) and two Fock spaces H ‘ , (n= 1,2), both of which are constructed from L 2 ( R 3 ) :‘H = L2(R)@’H1@’H2. The spaces ‘Hn(n= 1,2) describe the two reservoirs and L 2 ( R )describes the junction between the reservoirs. In terms of standard annihilation operators a, an,k satisfying canonical commutation relations: [‘&at]= 1,

[ a n , k ,a A t , k , ] = sn,n’d(k - k’) 7

(2.1)

the Hamiltonian is formally given by

where (h.c.) means Hermitian conjugate, W k = lkI2/2, k-integrations are taken over R3 and gn(k) E L 2 ( R 3 )( n= 1,2). Strictly speaking, the free field parts dkwkaL,kan,k (n = 1,2) should be understood as the second quantization of the multiplicative self-adjoint operator p(k) + wkp(k) defined on L 2 ( R 3 )to the Fock spaces ‘Hn (n= 1,2), respectively. The Hilbert space ‘H is a Boson Fock space over a Hilbert space

s

equipped with the inner product

The CCR algebra A, which corresponds to a set of observables, is generated by Weyl operators (cf. Theorem 5.2.8 of [2] ):

W(f) = exP (i@(f))

(2.5) where @(f) = @s(f) @ b , n ( f ) is a linear map from the Hilbert space L 2 ( 3 f ) to the space (3 @(f)) of unbounded operators on 7-l with

+ xn=1,2

In the above, the overline stands for the closure. On A, the Hamiltonian H generates a time evolution automorphism Tt : A + A. In order to specify the states over A, it is enough to investigate the average values of the Weyl operators. Indeed, any element of A can be approximated with arbitrary precision by a finite linear combination of finite products of Weyl operators. And a product of Weyl operators reduces

212

to a single Weyl operator because of the identity (Theorem 5.2.4 of [2]): W.f)W(g) = W(f 3- 9)exp (-W.f, 9)). We assume that the two fields and the junction are initially decoupled and, in the initial states W O , the two reservoirs are in equilibrium with different temperatures pF1, and densities p l ,p2. Nonequlibrium steady states are constructed as t -+ 03 limits of the state wo o ~t at time t. Since we have studied the case where both initial reservoir states are normal14, we consider the case where at least one of the initial reservoir states has condensate. Equilibrium states with Bose-Einstein condensate were firstly constructed by Araki-Woods2' and were investgated further by Cannon21 and LewisPu1&22.In these approaches, the states are prepared by taking an appropriate infinite volume limit. Starting from a finite cubic box of volume V where N particles are contained and obey the periodic boundary condition, the canonical average of the Weyl operator is shown to converge to a well-defined limit in a thermodynamic limit where V -+00 and N -+ 00 while keeping the density p = N / V constant20>21'22.When p exceeds a critical value, the condensate appears and the thermodynamic limit of the average of the Weyl operator acquires a factor represented by the 0th order Bessel function Jo(z). With the aid of the formula Jo(z) = J'T, g e i z s i n ( e + athe ) , thermodynamic limit of the canonical state is shown to be decomposed into gauge noninvariant pure equilibrium states where the phase of the condensate wave function is fixed. According to these results, the average of the reservoir Weyl operator at an initial state wo can be taken as

When the density pn is less than the critical value pc(,L?n)

On(+,)

=0

/ &epn2k-I, ,

where the chemical potential p, is a unique solution of 1 Pn = &ePn(y*-Pn) -1 *

J

On the other hand, when pn 2 pc(Pn), one has

an

where _= ,on -pc(,&) and arg are the density and the phase of the condensate, respectively. Note that (2.6) is meaningful only for Weyl operators, where +,(k) (n= 1 , 2 ) are continuous.

213

3

Nonequilibrium Steady States

To construct NESS as t

-+

00

limits of w o o q,we assume the followings:

(A) The initial state satisfies (wo(ab1abz . .

.&)I 5 n!K,

(3.1)

where abj = a or a+ and IS,(> 0) satisfies limn+m K,+l/K, = 0.

(B) The form factor g n ( k ) is a continuous function of (kJand C?(R3). Then

is in L1(R+)and uniformly Holder continuous with index 1/2.

(C) There exists no real solution for q ( z ) = 0, where (3.3) and l / q - ( w )

= l / q ( w - i0) is bounded.

The assumptions (A) and (B) are posed to simplify the investigation, while (C) plays an essential role as it guarantees the existence of the steady states. As mentioned before, time evolution of the states is fully specified by the behavior of the Weyl operators wg o Tt(W(f)). Since the Hilbert space 3-1 is a Boson Fock space over the Hilbert space 12, the bilinear Hamiltonian H is the second quantization of the Hamiltonian h densely defined on k?:

hf=h(&)

=

(

flc +

.f

Cn=1,2 dkgn(k)*&(k) "-'k$l(k) -k Agl(k)C

wk$'Z(k)

+ Ag2(k)c

1.

(3.4)

The group Tt of time-evolution automorphisms generated by H satisfies (cf. the argument before Proposition 5.2.27 of [2] )

(W(.f))= W ( e Y ) and, under the condition (C), one has Tt

214

7

(3.5)

Then, since the two fields and the junction system are independent at the initial state, the average value of the Weyl operator at time t is evaluated as

n

wo 0 Tt ( ~ ( f )=)wo ( ~ X P( i + ~ s ( e Z ~ ~ f ) ) ) wo (exp (i+b,n(eihtf))) (3.7) n=1,2

By evaluating each factor of the above equation, one obtains

Theorem 3.1 For the Weyl operator where each & ( k ) is Cr(R3),we have

where A ( f ) and A(f) are defined by

(3.10)

This implies that the steady state

w+

exists and that it is quasi-free with (3.12)

I n the above, the convention +n = 0 f o r pn 5 p c ( p n ) i s used. Under the time evolution T t , mass is conserved. Namely, one has (3.14) where Jn ( n = 1 , 2 ) are mass flows from the reservoirs: Jn

= iX/ dk ( g n ( k ) a A , k a - gn(k)*atUn,k)

And, as a straightforward corollary, one has

215

(3.15)

Corollary 3.2 The average mass flow at NESS is a s u m of the normal and Josephson currents:

The rest of this section is devoted to the proof o f the proposition. Firstly, by the same arguments as in [14] , one can show lim wo (exp (i+.s(eihtf))) = 1

t-00

Next we consider the quadratic part of log wo (exp ( i @ b , n ( e i h t f ) ) ) :

where

The time-dependent terms IL”(t) (n = 1,2 : j = 1,2) are shown to vanish in the limit o f t + +co with the aid of the following Lemma. Lemma 3.3 Let W ( W ) be uniformly Holder continuous with index 1/2, i.e., ]v(w’) - v(w)l 5 Klw’ - w11/2, v(0) = 0, suppv c [0,R],then the integral @I)

J(w,t)= satisfies, for an arbitrary R’ > 0,

dw’

,i (w’ - w ) t

w - WI - io

sup J J ( w , t ) J-+ 0 (t + +co) O
Indeed, I ( w ;t ) can be rewritten as v ( w l ) e i ( w ’ - w) t

00

I ( w ; t )=

dw‘

216

w - w’ - io

.

where w(w’)satisfies all the assumptions of Lemma because g n E Cr(R3)and the uniformly Holder continuity ofl/q+(w) and d&,(k, f ) . Thus,

s

lim ~-+CC

sup II(w;t)l = 0 OswSR‘

On the other hand, in terms of u,(w) = g;(&)

(‘R’ > 0)

(3.17)

[ J d i cp,(k, f)]

Ikl=&

and

R’ such that supp gz c [0, R ’ ] , one has

which leads to

The integral converges as the integrand may have, at most, a 1/&-singularity at the origin and, thus, I,?’(t) -+ 0 as t + 00. Similarly, limt+cCI,(2) ( t )= 0. Finally, we consider the linear part of logwo (exp ( i @ h , , ( e i h t f ) ) ):

where

En,

and G(W’) = s;, o+(w,) (rn) [ ~ d k ’ c p n , ( k ’ , f ) l l kis, =continuous. ~ As 47 is integrable, Riemann-Lebesgue theorem23 implies S$J,(O, t ) + 0 (t -+ GO). 4

Boson on Lattice

In this section, we consider the free Boson on the three dimensional lattice Z3. Let aj and a; ( j in Z”) be the creation and annihilation operators satisfying [ a j ,a;] = 6jk. Smeared Boson operators, a * ( f ) and a ( f ) ,are defined as

217

where f j is a rapidly decreasing function on Z3. fj is rapidly decreasing if and only if its Fourier transform f(k) is smooth( infinitely many differentiable). Set

We denote d ( Z 3 ) by the Weyl CCR C*-algebra generated by W(f). Next we consider equilibrium states above the critical density. For our purpose, it is convenient to consider the Bosons on a general infinite graph. Let I? = { V , E } be a graph where V is the set of vertices and E is the set of edges. We suppose that the Graph r is connected and it is a union of finite connected graphs rn , r = UF=lFn. The standard discrete Laplacian hr on r is defined by

where the sum k j is taken for all vertices Ic connected to j by an edge of and c ( j ) is the number of edges connected to j. The discrete Laplacian hr, for the finite sub-graph is defined in the same manner. If c ( j ) is bounded as a function on vertices hr is a bounded selfadjoint operator on Z2(r).When the graph is finite, the constant function 1 is the eigenvector for the smallest eigenvalue. h r l = 0. Furthermore, the Perron Frobenius theorem for positive matrices implies that hr is a positive operator and the constant function 1 is the unique(up to multiplicative constant) ground state for hr. The free Hamiltonian for Boson on r is the second quantization of the standard Laplacian hr:

Hr

= 'jJc(j)a;aj

- 1/2

C(a;a,

+a ~ a j ) ) .

(4.3)

k+j

ja-

Then,

[Hr,a*(f)l= a*(hrf) We can introduce the rapidly deceasing functions for arbitrary graph by using the distance of the graph. (We omit the detail.) The Weyl CCR C*-algebra on I? can be introduce as before. Then, the time evolution of an observable Q in the Weyl CCR C*-algebra d(r)on r is determined by

In particular,

218

rn,

For each finite sub-graph let vL!’) be the equilibrium state at the inverse temperature ,B and the density p:

(4.4) where the fugacity z , (0 < zn

< 1 ) is determined by

xjcr

f j = 0). When r is a finite graph, We define l ; ( r ) = {( f j ) E 12(r) 1 EOwill be the projection from 12(r) to l;(r).We set Tro(Q)= T r ( E o Q E 0 ) . Proposition 4.1 (i) Suppose that

Then, for any p, there exists z , such that limn z, = z , < 1 and

(ii) Suppose that the following limit exists: (4.8)

If p < @), limn z , If P 2 p(P),

= z,

< 1 and the equations (4.6) and (4.7) are valid.

limz, = 1 , lim n

n

zn Irnl(l- zn) = P - p w

(4.9)

Furthermore, we assume that any f = ( f j ) with compact support is in the domain of h,’/2. Then, any rapidly decreasing f and g are in the domain of h,‘/2 (hence in the form domain of 1- e-phr) and py’P)(W(f)) = li&fP)(W(f)) 1 = exP(-2(P

1

1+e-Dhr

-/I(P))lxr(f)t2)exp(-i(f’1 - e-phr f ) )

219

(4.10)

(4.11)

where (4.12) The Bose condensation occurs for the second case of the above proposition. Due to the equation (4.10) and (4.11), we have off-diagonal long range oder and the U ( l ) gauge symmetry breaking for the state ‘pk’”) with high density. The decomposition of into factor states (pure phase) is as follows. Set

+ip’e’(W(f)) = expji(p - p ( P ) ) 1 / 2 ( e i e x r ( f+ ) e-zeXr(f>)I (4.13) Then, (4.14)

The proof is same as that for the Newmean boundary condition in the continuous space R3. See[2].

Now we turn to the integer lattice. As we can compute eigenfunctions of discrete Laplacians on r = Z 3 as well as half infinite lattices, it is easy to see that Bose condensation occurs. The spectrum of the discrete Laplacians for r = Z 3 and half infinite cases are same but eigenfunctions are slightly different. Proposition 4.2 Set r
{ j = (jl,j’L?,j3)E z31jl 5 N } , r N < = { j = ( j l , j 2 , j 3 ) E Z 3i N 5 jl}, (4.16)

r = z3, rR= rol, rL=

O n these lattices, the Bose condensation occurs, namely, all the assumptions of Proposition 4.1 (ii) are valid.

220

5

NESS

Next we consider NESS(non-equilibrium steady states) and the current. The Hamiltonian we consider is bilinear in creation and annihilation operators but it contains interaction with inpurity or with different particles in the middle layer. More precisely, we consider the following situations. (a) The Initial state is the product of the condensate states on the left infinite lattice l?L and the right r R . The time evolution is the discrete Laplacian which allows Boson hopping from left to right and vice versa. (b) We introduce interaction with impurity in the middle. (c) The lattice is composed of three different parts, and the interaction between the middle layer and left and right infinite parts translationally invariant in the tangential direction of the junction. In all of these cases the total space is the three dimensional lattice r = Z3 (a) Free Boson. As before we set rR = ro<,r L = l?<-l, r = Z3 = l?L U ~ R The . Weyl CCR C*-algebra A = d(r)is the tensor product of the left and right Weyl CCR C*-algebras, d R = d ( r R ) and AL == A(rL), A = AL 63 d R . The Hamiltonians we consider here are

H L = Hr, , HR = HrR , HF = Hr hL = h p , , hR = hrR , h F = hr. Thus the Hamiltonian H F allows hopping from left to right and vice versa. The initial state 90 is the product state of condensed states for H L and HR. ‘PO = CPL63 ‘PR ,

‘PL = G ( p L , p L , o L )

‘PR = 4 ( p R , p R , o R )

?JIL!c’eL*R)

(5.1)

where the ‘ p and ~ p~ are the factor states on r L , R introduced in (4.13). Theorem 5.1 For any rapidly decreasing f o n Z3 the following limit exists: t-w lim

‘Po(n(W(f))) = ‘Pw(W(f))

J--i

=e x p { T [ e

i8L

(PL

x exp{-q(f, 1 (1 +.-‘)(I

-f(pL))1’2

+ eisR(PR - @ R ) ) ” 2 ) X r ( f )

- e- B

-1

f))

-k C.C.]}

(5.2)

where C.C. refers to the complex conjugate and H is the multaplication operator o n L2(T3)(via Fourier transform) defined by : p ~ w ( I cf) (Ic), p R b J ( k ) f(k),

221

when -7r < Icl < 0, when 0 < k l < 7r,

(5.3)

w(k) = 6 - 2 ( ~ 0 ~ +k Cl

O S ~+~c

os~~)

(5.4)

To show (5.2) we use the stationary phase method and the following formulae;

Next we consider the particle current from right to left. As is in other nonequilibrium steady states we define the current via the following procedure. First we consider the finite volume Hamiltonian Hr(N) defined by r(N) = { j =

(jl,j 2 , j 3 ) E

rl

Ijil

5 N 1 l j z l 5 N 1 1331 I N),

and the number operator,

QZ = C

agaj

, FaN) = (0 I ji I N,Ij2l I N , Ij3l 5 I N )

jEraN)

The current from right per area (of junction surface) j~ is defined by

Then, by (5.2),

In particular the current vanishes when ,LIL = PR, and the equation (5.6) reads that no effect of condensation appears in the current in bulk.

(b) Impurity Scattering First fu positive integers N , M , and we set

r = z3 = r<-M u r s u r N -< rs = { j = ( j i , j 2 , j 3 )

E Z31 - M

+ 1I j 1 5 N

-

1)

Thus we consider the system composed of three parts, two half infinite (condensed) Boson reservoirs on r <-- M and r N <- and two dimensional layer rs

222

between these reservoirs and we suppose that the intermediate layer rs contains impurity. The total Hamiltonian H consist of the hopping term and the impurity term as follows

H ( X ) = HF +

(5.7)

Xka;ak. kErs

The time evolution rt is determined as before, r t ( Q ) = eitH(X)Qe--itH(X) For simplicity we assume that ( X k ( is finite. At the level of the one-particle space l 2 (Z3), the above Hamiltonian corresponds to the following Schrodinger operator:

xkErs

h ( X ) ( )(k) f = h F f (k)+ X k f (k).

(5.8)

( We set X I , = 0 if k is not in rs.) It turns out that h(X)of (5.8) is a trace class perturbation of h H if l X k l is finite, and the following wave operator v exists on f in the absolute continuous subspace of h(X);

xkErs lim

e--ithFeith(X)

f =Vf.

(5.9)

t-LX

As the initial state, we consider the product of two condensed states on r 5 - M 1 and the equilibrium state on I'S associated with the discrete Hamiltonian hs, the inverse temperature PS and the the density ps. (Note that the condensation cannot occur on rs due to two dimensionality.) rN<

PO = PL 8 PS 8 P R ~

,

PL = V ! J ( ~ ~ L , ~ L , O PR L ) = V!J(DR,~R,OR)

xkErs

(5.10)

Theorem 5.2 W e assume that IXkl is finite and the Schrodinger operator does not have point spectrum. Then, the wave operator V in (5.9) exists on any f in 1 2 ( Z 3 ) and it is unitary. For any rapidly decreasing f on z3,Vf is in 11(Z3). Let cp be any state quasi equivalent to cpo of (5.10). The following limit exists for any rapidly decreasing f o n Z3 : lim P(Tt(W(f))) = V)m(W(f))

t-m

1 e-v*Hv)(l- e - V * f i V ) - l f ) } x exp{-q(fl (1+ We focus on the case when P = PL = PR , p the equation (5.11) has the following form:

223

= p~ = ,OR, and QL

(5.11)

# OR.

Then

(5.12)

+

The phase factor (1 e i ( e L - e R ) ( 2 is consistent with the Josephson formula which tells us that the current is proportional to a function of the phase difference (3L - OR. The total current J R from r N < defined by

contains a non-vanishing term proportional to 11 the 1st order perturbation theory.

+

ei(eL-eR))2

at the level of

( c ) Two-Dimensional Layer Next we proceed one step further. The interaction between the intermediate layer and the reservoirs is tranlationally invariant in the direction tangent to the junction. We divide the three dimensional lattices into three parts as before, I? = Z3 = r 5 - M U rs U r N g . The Hamiltonian is

H(X)= HF

+

c

+a*Wfd).

(a*(frc)arc

(5.13)

kErs

The real constant X is the coupling constant and the smearing function fk is called the form factor. We assume that the Fourier transform of f k is smooth and f(kl,kzlk3)(jl,jZ,j3)

= f ( k 1 , 0 , 0 ) ( h , j 2- b , j 3 - h)

(5.14)

for any intergers IC2 and k 3 . This condition (5.14) implies that the total Hamiltonian H(X) is translationally invariant in the 2nd and 3rd directions and we say H ( X ) is translationally invariant in the vertical direction. The corresponding Schrodinger operator is denoted by h(X) = h F

+ XK

(5.15)

The interaction K is no longer of trace class, however, due to translational invariance (in the vertical direction), the Hamiltonian h(X) is simultaneously

224

diagonalizable with the vertical shift operators on Z3. As a consequence, the Hamiltonian h(X) is the direct integral of a smooth family of one-dimensional discrete Schrodinger operators with finite rank interaction. This observation shows the asymptotic completeness of the wave operator for the pair h(X) and h F and the absence of singular continuous spectrum for h(X). For our study of condensed states, we require the 11(Z3) convergence of the wave operator. Theorem 5.3 W e assume the following T 2 condition for the interaction K in (5.15): (5.16)

f o r any rapidly decreasing f. Assume further that h(X) does not have any point spectrum. Then, the wave operator V = limt e-ithFeith(X) maps the set of rapidly decreasing functions to l ’ ( Z 3 ) and all other conclusion of Theorem 5.2 i s valid. We have several examples of the form factor fk for which the T 2 condition (5.16) can be verified. For example if N 5 M = 1 f(-l,o,o) = f(l,o,o) = -2f(0,0,0) and the Fourier transform f(o,o,o)(Ic)of f(o,o,o) vanishes in a neighborhood of the origin, the T 2 condition can be verified. We can consider the current j~ per unit area as defined in (5.5). Unfortunately, for all the examples satisfying the T 2 condition which we are aware of, the current vanishes. Our physical interpretation of vanishing current in bulk is as follows. The T 2 condition is an infrared cut-off for the form factor and the infra-particles can not interact under the T 2 condition. As a result, there is no friction in condensed states and the current vanishes. Acknowledgments One of the authors (ST) thank Professor T. Hida for his invitation and hospitality at “International Conference on Stocahstic Analysis; Classical and Quantum” (Meijo University, 1-5 November, 2004). This work is partially supported by Grant-in-Aid for Scientific Research (C) from the Japan Society of the Promotion of Science, by a Grant-in-Aid for Scientific Research of Priority Areas “Control of Molecules in Intense Laser Fields” and the 21st Century COE Program at Waseda University “Holistic Research and Education Center for Physics of Self-organization Systems” both from the Ministry of Education, Culture, Sports, Science and Technology of Japan and by Waseda University Grant for Special Research Projects, Individual Research (2004A-161).

225

Appendix

A

Proof of Lemma

Let

R1

= 2 max(R, R’) = Zk, then one has

-

J ( w , t ) = -J(w,t) - V(W)

I”’

ei(w‘-W)t

dw’ w ’ - w + i o

By changing the contour from [0, R1] to [0,i] U [i,R1 +i]U [R1+i, R I ] the , second term of (A.l) is shown to vanish as t -+ 03 uniformly for w E (0,k]. On the other hand, one has

It is easy to show that the second and third terms are bounded by 4 K f i and, thus, vanish as t

+ 00.

Because of

K G ( ~ ~ + ( W ’ - W Iw’

- WI

Iw‘

- w + $1

1/2

andfl+l~’-w+~1~/~51+(R+R’+l)~/~=K’fort>lr,we

and, thus,

226

where the integral in the last line can be easily seen t o converge. Therefore, J ( w , t ) converges t o 0 as t --+ +cm uniformly for w E [0, E].

References 1. 0. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics vol.1 (Springer, New York, 1987). 2. 0. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics v01.2, (Springer, New York, 1997). 3. H. Spohn and J.L. Lebowitz, Commun. Math. Phys. 54, 97 (1977) and references therein. 4. J. Bafaluy and J.M. Rubi, Physica A153, 129 (1988); ibid. 153, 147 (1988). 5. J. Farmer, S. Goldstein and E.R. Speer, J . Stat. Phys. 34,263 (1984). 6. J.-P. Eckmann, C.-A. Pillet and L. Rey-Bellet, Commun. Math. Phys. 201, 657 (1999); J. Stat. Phys. 95,305 (1999); L. Rey-Bellet and L.E. Thomas, Commun. Math. Phys. 215,1 (2000) and references therein. 7. T.G. Ho and H. Araki, Proc. Steklov Math. Institute 228,(2000) 191. 8. S. Tasaki, Chaos, Solitons and Fractals 12 2657 (2001); in Quantum Information 111, eds. T. Hida and K. Saito, 157 (World Scientific, Singapore,2001). 9. D. Ruelle, Comm. Math. Phys. 224 , 3 (2001). 10. S. Tasaki, T. Matsui, in Fundamental Aspects of Quantum Physics eds. L. Accardi, S. Tasaki, World Scientific, (2003) p.100. 11. V. JakBiE, C.-A. Pillet, Commun. Math. Phys. 226, 131 (2002). 12. J. F’rohlich, M. Merkli, D. Ueltschi, Ann. Hen& Poincare‘ 4,897 (2003). 13. J. Lauwers, A. Verbeure, A microscopic model for Josephson currents, J. Phys. A : Math. Gen.37 3577 (2004) 14. S. Tasaki, L. Accardi, submitted (2004). 15. I. Ojima, H. Hasegawa and M. Ichiyanagi, J. Stat. Phys. 50,633 (1988). 16. I. Ojima, J . Stat. Phys. 56,203 (1989); in Quantum Aspects of Optical Communications, (LNP 378, Springer,l991). 17. V. JakSiE and C.-A. Pillet, Commun. Math. Phys. 217,285 (2001). 18. V. JakSiE and C.-A. Pillet, J. Stat. Phys. 108,269 (2002). 19. D. Ruelle, J. Stat. Phys. 98, 57 (2000). 20. H. Araki, E. J. Woods, J . Math. Phys. 4,637 (1963). 21. J. T. Cannon, Commun. Math. Phys. 29,89 (1973). 22. J. T. Lewis, J. V. Pulit, Commun. Math. Phys. 36,1 (1974). 23. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, (1948).

227

MULTIDIMENSIONAL SKEW REFLECTED DIFFUSIONS GERALDTRUTNAU

Center for Interdisciplinary Research, Wellenberg 1, 33615 Bielefeld, Germany e-mail: [email protected]

Summary. Let G c Rd be (non-empty) open with euclidean closure # Rd. We assume that G is the interior of G. Let p E L1(Rd,dz), p > 0 dz-a.e., 0 5 a 5 1, and pa := ( C Y ~ G (1 - a ) l w d \ ~ ) pm, , := padx. Let D, = R d \ E i f a = 0, D, = Rd if 0 < a < 1, D, = G if a = 1. Let A = ( a i j ) be symmetric, and globally uniformly strictly elliptic, p

+

satisfying some mild regularity assumption. Non-symmetric conservative diffusion processes X t E D, which inside D, \ BG obey the generator L given in the following suggestive form d

d

xf=lso,

where p i l ( B 1 , ...,Bd) E L'(D,;@,m,) satisfies Biaifdz = o for all f E Cr(a,), are constructed and analyzed. The coefficients are in general neither continuous nor locally bounded. For dG, p, uij, sufficiently regular, a Skorokhod type decomposition for X t is given in Theorem 5.2. Indeed, due to the jump of pa along aG, the generator L has to be regarded with boundary conditions, or as integration by parts show, with additional surface measure terms. To these surface measures, there are uniquely associated multidimensional local times similarly to the case of reflected Brownian motion, where the local time is associated to the Dirac measure in zero. Under the irreducibility assumption of the semigroup corresponding to the symmetric part of L we show in Theorem 6.5 that X t is recurrent for q.e. starting point. Because of the non-symmetry ( a ,1 - a ) we can speak about skew reflection. The extreme cases are a = 1, a = 0. In particular, if a = there is no reflection.

4

228

The constructed process reminds one of a multidimensional generalized analogue of the a-skew Brownian motion (see [ 5 ] , [9]), which corresponds to the case d = 1,G = Rf,p = 1,Bi = 0, aij = 6ij. 2000 Mathematics Subject Classification:6OJ60,60555,31C15,31C25, 35525.

Key words: Diffusion processes, local time and additive functionals, potential and capacities, Dirichlet spaces, boundary value problems for second order elliptic operators.

1

The generalized Dirichlet form and its capacity

The following is an extension result corresponding to the diffusion constructed in [13]. Let G c Rd be (non-empty) open with euclidean closure # Wd. Since we are not interested in the geometry of G, we may assume that G is the interior of E. Then dG = d c = d(Rd\E). Let dx be the Lebesgue measure on Rd, p E L1(Rd,dz).Let 0 5 a 5 1, and put

D,

:=

{

Rd\E

fora=O

I W ~ f o r ~ < a <1 G fora=l.

We equip D, with the usual Euclidean norm 1 . 1 = Let p, = ( a l c p + (1 - a ) l w q c p )with p > 0 dx-a.e., and m, := padx. Let 1.11, 1.12, I . lo , denote respectively the L1-norm, L2-norm, L"-norm, w.r.t. m, on D,. For V c L1(D,,m,) let VI,:= DnLoo(D,,m,), V + := {f E Vlf 2 0 m, - a.e.}. Let A = ( a i j ) l s i , j g be measurable, symmetric, and uniformly globally strictly elliptic on D,, i.e. 3X > 0 such that

xt? C d

x-'

d

5

a= 1

+=I

d

I ACE? V(G, ...,t d ) E I K ~ ,ma-a.e. z E D,.

u i j ( z ) ~ j

(1)

i= 1

For k E N U {m}, let Ck(Rd) be the k-times continuously differentiable those functions in Ck(Rd) which have compact functions on Rd, and @(ad) support. For D c Rd closed let

C ( y ( D ):= { U l D : u E C,-(Rd)}. 229

Denote by D, the euclidean closure of D,. Let us from now on assume that €"(ti,

w) := 1 l , ( A V u , Vv) d m , ; u, w E CF(D,),

is closable in L2(D,,m,). Conditions in order to have closability are very mild. For instance, if p satisfies the Hamza type condition in D,, or &p E Lto,(Bd,d z ) , 1 5 i 5 d , then (E', is closable in L2(D,, ma).This follows easily by adapting arguments from [13]. Let us denote the closure of (€", Cr(D,)) on L2(D,, m,) by (E', D(E')). (E",D(€'))is a symmetric Dirichlet form (cf. [4]).(€',D(€")) needs not to be regular (compare [4,p.6] for the definition) on D,. Let for instance Q = 1, then Co(D,) n D(E') is in general not dense in D(E'). It becomes a regular Dirichlet form on with the identifications specified in the following remark.

Cr(n,))

n,

Remark 1.1 If 0 < Q < 1, then C0(B5,)no(€') is dense in D(€').Obviously, - CO(n,)nD(ET) is dense in Co(D,). Therefore (E', D(E"))is regular o n D , = D,. For Q = 0, or a = 1, we can make it a regular Dirichlet form on through the following identification. W e first extend m, to by setting m,(dG) := 0, thus identifying LP(D,, m,) with Lp(Da,ma),p 2 1. W e then may regard (E", D(€'))as a Dirichlet f o r m on L2(na,ma). I n this case (E", D(E')) is regular. Henceforth, we will make these identifications.

n5,

n,

Let ( G ; ) O , ~ be the resolvent associated to (€", D(€")) (see [4, Chapter 1.3]), and (., .) be the inner product in L2(D,, ma).For p > 0 let €;(., .) := E " ( . , . ) p(., .), I . ID(E') := E ; ( . , .)* be the norm corresponding to E'. Then D(€') is the completion of C?(Da) w.r.t. 1 . ID(^). We denote by (L",D ( L " ) ) the generator associated to ( E T ,D(E')), i.e.

+

D(L') = {f E D(E') : g H E ' ( f , g ) is continuous w.r.t. (.,.I* on D(E")), and for f E D(L"), L" f is the unique element in L2(D,,m,) such that ( - L " f , g ) m = € " ( f , g ) for all g E D(€"). Since p E L 1 ( R d , d z ) ,one can easily see that lg, E D(&"), and €'(lDa, f ) = 0 for any f E D(&"). Hence 16, E D ( L T )and L'16, = 0, i.e. L" is conservative. In particular Gila, = (1 - L')-'lna = (1 - L')-'(I - L')I- Dcv = 1-D,'

i.

Remark 1.2 Suppose that as in Theorem 5.2 below, everything is SU ciently smooth. Let A = ( Z i j ) I < i , j < d , where & j is a 4.c. version of aij. Let f E Cr(D,), Q # Then f is in D(L'), if

(XVf , 17) = 0 Tr(p)da-a.e. 230

-

Indeed, it sufices to look at the proof of Theorem 5.2. We consider a measurable vector field B : D, EXd, which is ma-square integrable on D,, i.e. JD IBI2dm, < CQ, and such that

LaCr(n,)

( B ,V u )dm, = 0 for all u E

Cr(n,).

(2)

Note, that since c D(&') densely w.r.t. I . I D ( ~ ) , and because of strict ellipticity (l),(2), extends to all of D(&').Furthermore,

( B ,Vw)udm, for all u,v E D(&')a.

L a ( B , V u ) v d m ,= -

(3)

Exactly as in [13,Proposition 1.41 we have the following:

+

Proposition 1.3 (i) The operator L'u ( B , V u ) ,u E D(L')b, is dissipative, hence in particular closable in L1(D,,m,). The closure (T,D ( z ) ) generates a sub-Markovian Co-semigroup of contractions (Tt)t>o.

(ii) D ( z ) b c D(E') and

In particular, E'(u, u)= -

La

LUU

-

dm, for ewenJ u E D(Z)b.

(5)

(iii) (Tt)t>O - is Markowian.

(Tt)t?ocan be restricted to a Co-semigroup (Tt)tlo on L2(D,,mQ). The corresponding generator ( L ,D ( L ) )is the part of D ( z ) )on L2(D,, m a ) , i.e. D ( L ) = {u E D ( Q n L 2 ( D u , m , ) l L E L2(D,,m,)}, and Lu := Eu, u E D ( L ) . Let (L',D(L'))be the adjoint of ( L , D ( L ) )in L2(Da,m,), and (T,')t>othe corresponding semigroup. According to [lo,Examples 4.9(ii)], the generalized Dirichlet form corresponding to ( L ,D ( L ) )is

(z,

E ( u , v ) :=

for u E D ( L ) , v E L2(D,,m,) for v E D (L '), u E L2(D,,m,).

(-~u,v) { (-L'v,u)

Define &p(., .) := &(.,

a)

+ p(., .). As in [13] one extends & as

&(u, w ) = I r ( u ,v) -

( B ,V u ) v d m , for every u,v E IDa

23 1

D(&')b.

(6)

As a consequence of (3) we then have

&'(u,u)= &(u,u) for every u E D(E')b.

(7)

(7) suggests that the capacities related to & and &' are equivalent. This will be shown further below. Let ( G p ) ~ > o(resp. (G&)p>o) the strongly continuous contraction resolvents on L2(D,,m,) related to ( L , D ( L ) )(resp. (L',D(L'))).As usual let fu denote the 1-reduced function o f f on U w.r.t. &. By abuse of notation (Gicp)~ will always denote the 1-coreduced of Gicp on U.Let e 7 u , f E D(E'), denote the 1-reduced function of f on U . If U = O, we simply write e; -

instead of e?"", and e f instead of f B m . Let Cap be the capacity associated to E' as defined in [4,p.641. Recall that of closed subsets of is called an (€'-) an increasing sequence (Fk)kE~ nest o n D,, if limk,, Cap(Fz) = 0.

n,

We fix throughout cp E L1(D,, ma),0 to & is determined by

< cp 5 1. The cp-capacity related

Cap,(U) = E1(Glcp,(Gicp)~)for U

c O,, U open.

An increasing sequence of closed subsets ( F k ) k > l is called an E-nest, if limk,, Cap,(Fz) = 0. A subset N c D, is cafied &-exceptional if there is an &-nest (Fk)kllsuch that N C nk>lD, \ Fk. The &'-exceptional sets are defined similarly.

Theorem 1.4 (a) ( F k ) k E ~ is an &'-nest o n iJ andonly if (Fk)kE~ is an E-nest.

a, in the sense of [4,p. 6'4

n,

(ii) A subset N of is &-exceptional iJ and only i f it is exceptional w.r.t. E' in the sense of [4,p. 1341. Proof It is enough to show (i). Let ( F k ) k E ~be an &'-nest. Then (see -+ 0 in D(E'). Since Glcp 5 lDait follows e:; + 0 in D(E'). [4]) er'FkC li% For

p > 0 let (Gicp& be the unique solution f

E D(L') to (1 - L ' ) f =

is known that (Gicp)& t (G',cp)~; and strongly in L2(D,, m,). It is easy to see that (G',(P)$~also converges weakly in D(E")

P(f -

~ F , c ) - .It

232

to (Gicp)~; as p

t 00.

Then using in particular (1)

It follows that (F) is an E-nest Converesely, suppose that (F) is an

where (G) is the resolvent associated to E Since

we obtain

and consequently k+oo

0 Having shown that E' and & have equivalent capacities, we can pronounce exceptional, quasi-continuous, quasi-everywhere, etc. , unambiguously and without specifying whether it is meant w.r.t. Er or &, and so we will do.

2

Construction of an associated diffusion

The results obtained in [13] are derived by using the following ingredients: strict ellipticity (l),finiteness of the reference measure ma,and m,-square 233

integrability of the vectorfield B.This is the reason why the results of [13] carry over to our situation. We will list below the results corresponding to [13]. The interested reader may then refer to [13]. In order to show the existence of an associated process we have to check quasi-regularity of & and condition D3 of [lo,IV.21. Doing the identifications of Remark 1.1, the quasi-regularity of the generalized Dirichlet form E on D, is in view of Theorem 1.4 a consequence of the regularity of the classical Dirichlet form E’ on D,. Since exactly as in [13]y := D ( z ) bis a linear space such that Y c L”(D,,m,), Y n D ( L )c D ( L ) dense, limp,, ef-oGpf = 0 in L2(Darm,) for all f E y , and such that f A p E y(=the closure of y in Loo(D,, m a ) )i f f E 7, p 2 0, the existence of an associated process follows from a general result in [lo].In particular one also shows that D ( r ) b is an algebra. Finally, we can also show that E is local in the sense of generalized Dirichlet forms and obtain the following. Theorem 2.1 There exists a Hunt process M = (0,(.F)t>o, (Xt)tzo,( P z ) z such that the resolvent Rp f ( x ) := Ez[~ow e-pt f ( X t ) d t ]is a quasi-continuous ma-version of G p f for any f E L2(D,,ma)b, p > 0 . I n particular M is conservative, i.e. P R p l ~ ~ (=x 1) for quasi every x E Dff,and there exists an exceptional set N c such that

P, (t H X , is continuous on

[o, m[) = 1 f o r every x E D, \ N .

Remark 2.2 Since -B satisfies the same assumptions as B,exactly as we have co_structed the process M associated to L one can construct the coprocess M associated to L’. The coprocess will have exactly the same properties than M.

3

The Revuz correspondence

(n,,l?(nff))

Definition 3.1 A positive measure v on charging no exceptional set is called smooth if there exists a nest ( F k ) k E N of compact subsets of D,,such that < m for all k E N.

The smooth measures are denoted by S . The positive continuous additive functionals (PCAF’s) of M are defined as in [12].The following theorem accomplishes together with [12, Theorem 3.11 the so-called Revuz correspondence for the generalized Dirichlet form E . It can be shown directly and exactly as in [13] (please see also the following Remark 3.3). 234

Theorem 3.2 Let u E S . Then there exists one, and only one PCAF ( A t ) t l o of M such that for any positive, Bore1 measurable f , and any quasi-continuous ma-version i7 of v E pb = {u E L2(D,, m a ) b I PG&+lu I u 'd,B 2 0}, we have h

s,, -

v f d u = lim ,B/ p+Oo

i7(x)Ex D,

[lW

e - ( p + ' ) t f ( X t ) d A t ] m,(dx).

(8)

I n this case we write u = V A . Remark 3.3 A consequence of Remark 2.2 is that the process M is in weak duality with the coprocess M. I n particular m, is an excessive measure for the process M. Theorem 3.2 could then also follow from results in [6, $91 where properties of Revuz measures are discussed in the weak duality context. The interested reader is invited to consult the given reference, and to compare carefully the fine (process) topology of [6], with our analytic capacity.

4

Semimartingale characterization

-

From [12,Theorem4.5(i)] we know that for any f E D ( L ) with quasicontinuous ma-version f we have a unique decomposition

ALfl := T ( X t )- T ( X 0 ) = Mif1+ Nif1, t 2 0,

(9)

where M [ f l is a martingale additive functional (MAF) of M of finite energy and N [ f l is a continuous additive functional (CAF) of M of zero energy. The energy of an additive functional A of M is defined by

e(A)

=

f hlP2

L, [I

00

Ex

1

e-OtA?dt m a ( d x )

whenever this limit exists in [O,m].The equality (9) is to be understood in the sense of equivalence of additive functionals 0,f M. If AIfl decomposes in the sense of (9) for some f with f E L2(D,, ma)we write f E L2(D,, ma)dec. Using the extension theorem [12,Theorem 4.5(ii)] it can be shown exactly as in [13] that D(E')b U D ( L ) C L2(D,, ma)dec.

Remark 4.1 For f E D(E') we obtain also the decomposition (9) except that we do not know whether e(NIf1)= 0 (cf. Remark 4.2 in [13]). If u E D ( L ) then clearly N/"] = s," L u ( X , ) d s is of bounded variation, i.e. it can be represented as the difference of two PCAF's of M. For general 235

u E D(&’)b we have roughly the following. Nt(ul is of bounded variation, if and only if there exist v1, vz E S, such that

for “enough”quasi-continuous w E D(&’)b. In this case Nt(ul = A: - A t , where A’, A’, are the PCAF’s associated with v1, vz. For the precise meaning of “enough”we refer to Theorem 4.5. in [13].

5

Identification of the process

Throughout this section we assume that G c Wd is a bounded Lipschitz domain, i.e. G is open, bounded, and its boundary dG is locally the graph of a Lipschitz function. Let G be the (compact) closure of G in Wd equipped with the usual Euclideannorm 1.1 = (., .)‘I2.Let R c Wd open. Let H’*P(R), p E [l,co[,denote the classical Sobolev spaces of order one in LP(R,dx), i.e. H’J’(S2) := { u E LP(R,dx)ldiu E LP(R,dx),1 5 i 5 d } . We will give some kind of Skorokhod representation of Xt when p E H1>l(Rd), aij E D(E‘). Note that p E H1?l(Wd)implies the closability of (E‘, Cr(D,)) in L2(D,, ma). Let (T be the surface measure on dG. Recall that since G is a bounded Lipschitz domain there exists a bounded linear operator

T r : H1’”(G) LP(dG,a), --f

(11)

called the trace on dG, and T T f) ( = f on dG for any f E H1”(G)nC@). Furthermore, the weak Gauss-Green theorem holds, i.e. if f E H’il(G), 1 5 i I: d , then

S,

d i f dx = -

S,,

T r ( f ) q id o

where q = (71,...,q d ) is the inward normal of F on dG (see for instance [3]). Let v = (vl,...,vd) denote the inward normal of Rd \ G on dG. We have vi = -qi, 1 5 i 5 d. Let g E CF(D,), and f E Cr(Rd) such that f ID, = 9. Let pm E C r ( W d )such that pm 4 p in H1y’(Wd).Then by (11)

236

and Gauss-Green theorem F

=

-

lG

gTr(p)vido.

For p E H1>'(Rd)we show that the weighted surface measure Tr(p)da,or equivalently l{T,(,),o}da, on dG is smooth.The proof is similar to the corresponding proof in [13] but we include because of some subtle differences. Theorem 5.1 Let p E H1vl(Wd).Then Tr(p)daE S, In particular for any w E D ( E T ) a and quasi-continuous ma- version 6 we have -

lG

GTr(p)qid a = k di(wp)dx = -

ai(wp)dx =

1,

GTr(p)vida.

Proof If h E C1(G) it is well-known (see e.g. [3,p.134, 3. (** *), (***)I) that there exists a universal constant C depending only on the Lipschitz domain such that

for any p E [1,m[. Let us choose ( p k ) & N c C ~ ( Rwith ~ ) pk -+ p in H1>'(Rd)as k 4 m. By (11) p k --f T r ( p ) in L1(dG,a) as k 4 00. Let K c Da be a compact set. Let f E Cp(Da),f 2 1 everywhere on K . Then, using (14), Lebesgue's theorem, and the Cauchy-Schwarz inequality,

5 max(a, 1- a ) Assume Cap(K) = 0. By [4,Lemma 2.2.7(ii)]

237

where CK = { u E

CF(D,)).(.I

2 1 , V x E K } . Hence, there exists

(fn)nEN c CF(D,), fn(x) 2 1, for every n E N, x E K , such that IfnlD(EP)+ 0 as n + 00. Since normal contractions operate on D(E')

we may assume that supnENs~pZERd Ifn(x)( 5 C. Selecting a subsequence if necessary we may also assume that limn--rooI fnl = 0 ma-a.e., hence dxa.e. on D,. Suppose a # 1, then D , 3 G. Consequently, using Lebesgue's theorem we obtain

)imL IfnllVpldx

= 0,

and therefore JKnaG Tr(p)d a = 0. Since T r ( p ) d c ,as well as Cap are inner regular the first assertion now follows for a # 1. If a = 1, we choose a compact domain with smooth boundary such that c V and dGndV = 8. Then U := V \ G is a bounded Lipschitz domain which is contained in D,. Let K denote its surface measure. As before we can then show that T r ( p ) d n is smooth. Note that 1aGTr(p)dK, = T r ( p ) d o . Thus T r ( p ) d a is also smooth. The second assertion is clear by (13) and since T r ( p ) d a is finite and smooth as we just have shown.

c

v

0 We present here below the identification of the process in Theorem 2.1 for a special class of p, aij, dG.

Theorem 5.2 Let p E H 1 ? ' ( R d ) ,p > 0 dx-a.e. Let A = ( a i j ) satisfy ( 1 ) with aij E D(E'), and q.c. m,-versions Z i j , 1 5 i,j 5 d. Let 0= ( a i j ) I < i , j < d be the positive square root o f t h e m a t r i x A . Let B = p-l(B1,..., be a ma-square integrable vector field satisfying (2). Let G be a bounded Lipschitz domain. Let q = ( q l ,..., r ] d ) (resp. v = (v1,...,V d ) ) be the unit inward normal vector field of on dG (resp. of Rd \ G on dG). Let &q)k := x y , l a k j q j , (resp. A(v)k := c j d _ l a k j v j ) , 1 I k I d , be the inward normal of G (resp. Rd \ G) associated with A . The conservative diffusion X = ( X ' , ..., X d ) of Theorem 2.1 is a semimartingale and has the following Skorokhod decomposition for 1 5 k 5 d:

z1

-

-

w

= (z1,..., Z d ) E D,,where w = (w',..., w ~is )a d-dimensional standard BM starting from zero, ( l f ) t 2 0 denotes the unique PCAF associated to the weighted surface measure T r ( p ) d u E S through

t 2 0, P,-a.s. f o r q.e. z

238

Theorem 3.2. I n particular rt

Proof The coordinate functions pk(x1, ..., xd) := xk are not in D(E')b. But they are locally in in D(E')b, i.e. p k f E cr(D,) C D(Er)b for any f E Cr(E,). Let f E Cr(Ea),( M l f l ) , be the square bracket of Mjfl. Then an easy calculation gives that the energy measure of Mif1,i.e. the Revuz measure of ( M [ f l ) , is , P ( M [ f l )=

( A V f ,V f ) d m a .

Thus ( M [ f l ) = , sof(AVf(X,),V f ( X , ) ) d s by Theorem 3.2. Let w E D(ET)b. Then by the previous results of this section

Let fl E Cr(D,), f l = 1 on K ~ ( o:= ) { x E D, : 1x1 5 l } , 1 2 I. Using the above, Theorem 5.1, (lo), Theorem 3.2, we easily derive the decompo~ ~0 ~ sition (9) for p k f l . One can also easily see that ~ K ~ ( O ) P ( ~ [ ~ =

._ for any m 2 1. This further implies that M F k f i l= M[pkfml 'dt < t - C K I ( O ) C .inf{t > 0 : X t E Kl(0)c}.Since obviously A P f i l = AIPkfml t 'dt < - QKI(O)C, and o ~ ~ ( o 00,) by ~ letting 1 --+ 00 we get the identification of the process. Since by (10) N F k f i lis of bounded variation for any I , the process is a semimartingale. Of course l f = laGn{Tr(p)>O)(Xs)&$, t 2. 0, by Theorem 3.2, because l a ~ n { T , ( p ) > o } T r ( p ) d a= T r ( p ) d a since supp(a) c dG.

s,"

0

Remark 5.3 W e would like to describe shortly the decomposition of Theorem 5.2. The diffusion has a symmetric drijl part corresponding to the logarithmic derivative of p associated to the diffusion matrix A. It has a 239

purely non-symmetric part p;’B, and two reflection parts. (15) tells us that ef’ behaves like a multidimensional local time o n some part of the boundary, i.e. Cf’ only grows when X t meets the boundary d G at those points where Tr(p) > 0. At that time, X t is reflected at, or passes through (see Remark 6.6), d G in normal direction associated with A . Finally note that if cx = there is n o reflection.

6

Recurrence

Let f E L1(Da,m,)+. Then

is uniquely determined at least for ma-a.e. z E 0,. Recall that 1 ~ ED(L)b and p G p 1 ~ -= 1~ for any p > 0. Therefore

G16, = 00 ma-a.e. We will need the following Hopf’s maximal ergodic inequality (cf. [4, Lemma 1.5.21, [8, Lemma 1.5.41). It can be shown exactly as in the sectorial case.

Lemma 6.1 Let h E L1(D,,m,), supnL1G g h ( z ) > 0). Then

p > 0, and let Ep

:= {z E

B, I

n

Hopf ’s maximal ergodic inequality is essentially sufficient in order to prove the following recurrence lemma.

Lemma 6.2 Let f E L1(D,,m,)+. Then { G f = co) U {Gf = 0) = D,, and { f > 0) c {Gf = co}, up to an ma-negligible set. Proof For arbitrary f E L1(D,,m,)+, a > 0, we set h := lg- - af in (16). Since B := {Gf < co) = {GlB0 = c o ) n { G f < 00) c Ep for any ,B > 0, up to a ma-negligible set we obtain

00, N --f 00, we get for any N 2 1. Dividing by a, and letting a PGp f dm, = 0, as well as G pf dm, = 0. Letting in the first case p 4 00, and in the second p 4 0, we get

sB

sB

l{Gf
=0

a~

well as 240

l { ~ f <. Gf ~ }= 0.

l { ~f<~}= . G0 implies f ({Gf< m}n{Gf > O } ) c = {Gf= m}U{Gf = 0 ) = D, up to ma-negligible set. l { G f < o o } - f= 0 implies {f > 0) c { G f = m}. This concludes the proof. 0

Lemma 6.3 Let f E L1(Da,m,)+. Then: (i) { G f < m} is invariant, {Gf = m} is co-invariant, i.e. for any p t > 0, h E L2(Da,ma),we have

> 0,

and

(zi) l{Gf=oo}is excessive,

l{Gf
is co-excessive, and l { ~ f < ~l {} ,~ f = E

D(&'). (iii) Let u,v and

E

D(&')b. Set B := { G f <

&'(u,v) = & ' ( l g u , lgv)

00).

Then l g u , l g v E D(E')b,

+ &'(lgcu, lgcv).

(17)

In particular &'(lgU,V)

= &'(lgu, 1 g v ) = ET(u,l g v ) ,

and B , hence also BC,is an invariant set w.r.t. E' Lemma 1.6.1 .].

in the sense of

[4,

Proof (i) We may assume that f is bounded. Put B = { G f < m}, B" = {Gf= co}. Let h E L 2 ( D a , m a ) + .Then for any n 2 1

241

Thus 1gCT[(hl{Gf5n)) = 0, and therefore l g ~ T [ ' ( g l= ~ )0 for any g E L2(D,,m,). It follows (lgTt(lBch),g)= (h,IgcT;(1Bg))= 0 and therefore l g T t ( l B c h )= 0. We conclude that (i) holds. We now show (ii). By (i) ,f?Gplgc = lgePGplgcI l g c P G p l ~=~l g = ,and

E'(PGplgc,PGplgc) = E(PGpli+,PGplBc) I E ( P G ~ ~ { G 10,) ~ = ~=)0., Hence l g c E D(€') easily follows from [7, I. Lemma 2.121. The proof corresponding to 1~ works similarly. (iii) Let p ( M [ w l )be , the energy measure related to (MIzo]),,w E D(E')b. Suppose that w is constant pu(M[wl)-a.e. on some Bore1 set A. Then it is well-known that J A d y ( M ( w l )= 0 (see [ l l ,Lemma 3.8. (iii)], but also [l, Corollaire 61, [2, equation ( S ) ] in the symmetric case). Note that p(M[wl) = (AVw, Vw) dm,. It follows that E T ( l g u l, g c v ) = E'(lg=u,l g v ) = 0. Indeed, by Cauchy-Schwarz

€ ' ( l g c u ,l g v ) = 0 follows in the same way. Thus €'(u,v) = € ' ( ~ B u1gw)+ , E T ( l g C u , l g C vReplacing ). first u by l g u , and then v by l g v in (17),the second assertion of (iii) follows. For the last assertion, exactly as in the proof of [4,Theorem 1.6.1.1, we use the second assertion of (iii) in order to see that for any h,g E L2(D,,m,)b,

From this, the invariance of B follows immediately.

0 Proposition 6.4 Suppose I' is irreducible (an the sense of (4,p . 481). Then Gf = m ma-a,e. for any f E L1(D,,m,)+ with ma({f > 0)) > 0. Proof By our assumption J D , f d m , > 0. Since { f > 0) c {Gf = m} by Lemma 6.2, it follows f dm, = fl{Gf=..) dm,, hence m,({Gf = m}) > 0. Since {Gf = m} is €'-invariant by Lemma 6.3 (iii), and E' is irreducible, we must have m,({Gf = 0 0 ) ~ ) = 0. Therefore the assertion follows.

so,

,,s

0 242

Theorem 6.5 Suppose E' is irreducible (in the sense of [4, p. 481). For r > o let ~ ' ( z ):= { y E D, : 111: - yI < r } , and c D r ( , ) := inf{t > OlXt E D T ( z ) } Let . (6t)t>0 be the shift operator correponding t o X t . T h e n P , ( ~ D ~ 0(6, ~ )< 00, ~n 2 0) = 1

for q.e. z

E

0,.

(18)

Proof The proof is similar to the proof of Theorem 4.6.6.(ii) in [4],but we include it in order to point excactly out the subtle differences. Let B c 0, be an arbitrary Bore1 set, and ( p t ) t 2 0 the transition semigroup of X t . The Markov property implies that f(z) := P,((TB < m) is excessive, since p t f ( z ) = P,(CJB0 6t t < m) 5 P,(oB < m). In particular due to the boundedness of f , standard arguments then imply that f E D ( E T ) b , and f is q.c. On the other hand for any positive g E CF(D,) with g dm, > 0 we have G'g = limp,o Gbg = 00 ma-a.e. Indeed, this follows immediately from Remark 2.2 and the co-version of Proposition 6.4. Then, using the resolvent equation

+

sD

0

5 (G&g,f-aRaf)= (g,Rpf-aRpRaf)= (g1Raf-PRpRa.f) 5 (9,Ra.f)< m.

Letting /3 0 we conclude aR, f = f , hence L f = 0 and thus f ( X t ) in (9) is a P,-martingale for every z E 0,\ N where N is some exceptional set. Let E := { f = l},y E [0,l),and E := { f = l},c-,:= c~~be the first hitting time of E-, := { f 2 y}. Note that E-, is finely closed. Now, for any z E E \ N, T > 0, we have by the optional sampling theorem ---f

1 = =

I

f (X)

= E z [ f(XuyAT)]

Ez[f(XuJ;e-/ I T I + E z [ f ( X T ) ; q>TI yP,(a-, 5 T ) Pz(a-, > T ) ,

+

which means that P,(e-,I T ) = 0. Thus

Pz(cp < m) = 0 for any z

EE

\ N,

and E is invariant w.r.t. E . Exactly as in the proof of Lemma 6.3(iii) one shows that E is then invariant w.r.t. E'. Owing to the irreducibility of E' we must have either m,(E) = 0 or m,(EC) = 0. Finally we let B = D T ( z ) . Then E 3 D T ( z )and m,(E) 2 ma(DT(z)) > 0, thus m,(EC) = 0. It follows f(z) = 1 for ma-a.e. z. But f is quasi-continuous, and therefore f = 1 q.e. (18) now follows from the Markov property.

0 Remark 6.6 W e have seen in Theorem 6.5 that i f E' is irreducible, then X t is recurrent q.e. in the classical sense. In particular, i f additionally 0 < a < 1 one can conclude that the process passes infinitely often through dG. 243

References [11 Bouleau, N.: DBcomposition de 1’6nergie par niveau de potentiel, ThBorie du potentiel (Orsay, 1983), 149-172, Lecture Notes in Math., 1096, Springer, Berlin, 1984. [2] Bouleau, N., Hirsch, F.: Dirichlet forms and Analysis on Wiener space, Walter de Gruyter, Berlin, 1991. [3] Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions, Boca Raton: CRC Press 1992. 141 F’ukushima, M.,Oshima, Y., Takeda, M.: Dirichlet forms and Symmetric Markov processes. Berlin-New York: Walter de Gruyter 1994. [5] Harrison, J.M., Shepp, L.A.: On skew Brownian motion, Ann. Prob., Vol. 9, N0.2, 309-313, (1981). [6] Getoor, R.K., Sharpe, M.J. Naturality, standardness, and weak duality for Markov processes, Z. Wahrscheinlichkeitstheor.Verw. Geb. 67, 1-62 (1984). [7] Ma, Z.M., Rijckner, M.: Introduction to the Theory of (NonSymmetric) Dirichlet Forms. Berlin: Springer 1992.

[8] Oshima, Y.: Lectures on Dirichlet spaces, Universitat ErlangenNurnberg, (1988). [9] Revuz, D., Yor, M.: Continuous martingales and Brownian motion, Springer Verlag, (1999).

[lo] Stannat, W.: The theory of generalized Dirichlet forms and its applications in analysis and stochastics, Mem. Amer.Math. SOC.,142 (1999), no. 678. [11] Trutnau, G.: On a class of non-symmetric diffusions containing fully nonsymmetric distorted Brownian motions. Forum Math. 15 (2003), no. 3, 409-437. [12] Trutnau, G.: Stochastic calculus of generalized Dirichlet forms and applications to stochastic differential equations in infinite dimensions, Osaka J. Math. 37 (2000), 315-343. [13] Trutnau, G.: Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non-reflection part. Probab. Theory Related Fields 127 (2003), no. 4, 455-495.

244

ON QUANTUM MUTUAL TYPE ENTROPIES AND QUANTUM CAPACITY

NOBORU WATANABE Department of Information Sciences Tokyo University of Science Yam.azaki 2641, Noda City, Chiba 278-8510, Japan E-mail: [email protected] The mutual entropy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel. The (semi-classical) mutual entropies for classical input and quantum output were d e fined by several researchers. The fully quantum mutual entropy, which is called Ohya mutual entropy, for quantum input and output by using the relative entropy was defined by Ohya in 1983, and he extended it to general quantum systems by means of the relative entropy of Araki and Uhlmann. The capacity shows the ability of the information transmission of the channel, which is used as a measure for construction of channels. The fully quantum capacity is formulated by taking the supremum of Ohya mutual entropy with respect t o a certain subset of the initial state space. One of the most important theorems in quantum communication t h s ory is coding theorem. The quantum coding theorems are discussed by using the mutual entropy - type measures introduced in the studies of quantum information. In this paper, we compare with these mutual entropy-type measures in order to obtain most suitable one for discussing the information transmission for quantum communication processes.

There exist several different types of quantum mutual entropy. The classical mutual entropy was introduced by Shannon to discuss the information transmission from an input system to an output system. It denotes an amount of information correctly transmitted from the input system to the output system through a channel. Kolmogorov, Gelfand and Yaglom gave a measure theoretic expression of the mutual entropy by means of the relative entropy defined by Kullback and Leibler. The Shannon's expression of the mutual entropy is generalized to one for finite dimensional Ohya took the quantum (matrix) case by Holev~,Ingarden,Levitin~~~~~~. measure theoretic expression by (KYG) Kolmogorov-Gelfand-Yaglom and defined Ohya mutual entropy la by means of quantum relative entropy of Umegaki 37~13in 1983, he extended it 2o to general quantum systems by 245

using the relative entropy of Araki and Uhlmann 38. Recently Shor 36 and Bennet et a1 6,4 took the coherent entropy and defined the mutual type entropy to discuss a sort of coding theorem for communication. In this paper, we briefly review quantum channels. We briefly explain three type of quantum mutual type entopies and compare with these mutual types entropies in order to obtain most suitable measure to discuss the information transmission for quantum communication processes. Finaly, we show some results of quantum capacity defined by taking the supremum of Ohya mutual entropy according to the subsets of the input state space. The capacity means the ability of the information transmission of the channel, which is used as a measure for construction of channels. 21123124725128929326

1. Quantum channels In development of quantum information theory, the concept of channel has been played an important role. In particular, an attenuation channel introduced in l8 has been paid much attention in optical communication. A quantum channel is a map describing the state change from an initial system to a final system, mathematically. Let us consider the construction of the quantum channels. Let 3.11 , 'Fl2 be the complex separable Hilbert spaces of an input and an output systems, respectively, and let B (3.1k) be the set of all bounded linear operators on 3.1k. 6 (3.1k) is the set of all density operators on "rtk (Ic = 112): 6 (3.1k) { p E B ( ' F l k ) ; p 2 0, p = p * , t r p = 1) (1) A map A* from the input system to the output system is called a

(purely) quantum channel. (2) The quantum channel A* satisfying the affine property (i.e., h = 1 ( v h 2 0) A* (Ck& P k ) = XI, A* ( P k ) 1 v p k 6 (3.11)) is called a linear channel.

*

XI,

A map A from B (3.12) to B (3.11) is called the dual map of A* : 6 (3.11) 6 (3.12) if A satisfies trpA ( A )= trh* ( p ) A for any p E

G (3.11) and any A

E

-+

(1)

B ('Fl2).

(3) A* from 6 (3.11) to 6 ( X 2 ) is called a completely positive (CP) 246

channel if its dual map A satisfies n

j,k=l

for any n E N, any Bj E B ('HI) and any AI, E B ('H2)

I

One can consider the quantum channel for more general systems. The --input system denoted by (A,6 ( A ) )and an outputsystem - by (A,B ( A ) ) , where A (resp. 2)is a C*-algebra, 6(d)(resp. 6 ( A ) )is the set of all states on A (resp. 2). When some outside effects should be considered in a certain physical process such as noise and an effect of reservoir, it is convenient to extend the system A to A @ B, where B describes the outside system. In such cases, the concept of lifting introduced in is useful.

(4) A lifting from A to A @ B is a continuous map &* : 6 ( d )+ 6 ( d @ B ) .

(5) A lifting €* is linear if it is affine. (6) A lifting &* is nondemolition for a state E * ' p ( A @ I )= 'p(A) for any A E A.

(3) 'p

E

6 (A) if

This compound state (lifting) can be extensively used in the sequel sections. The concept of lifting came from the above cmpound state (nonlinear lifting) l8 and the dual of a transition expectation (linear lifting) l , hence it is a natural generalization of these concepts. Let (0,Sn, P (0)), (6,Sn,P

(6))

be input and output probability

fi) and P ( R ) , P (6) are the sets of regular probability measures on s2 and 6. A channel E* transmitted from a probability measure to a -quantum

spaces, where Sn (resp. 86) is a a-algebra of R (resp.

state is called a classical-quantum (CQ) channel, and a channel S* from a quantum state to a probability measure is called a quantum-classical (QC) channel. The capacity of both CQ and QC channels have been discussed in several papers A channel from the classical input system to the classical output system through CQ, Q and QC channels is now denoted by 9321i26.

(7

P ( R ) 5 6 ( ' H ) I f B 7? 3 P R

( ) =*

One of the examples of the CQ channel Z* is given in 247

29

as follows:

CQ channel: Let C(R) be the set of all continuous functions on 0. For each w E R, we assume that pw E 6 (3-1) is - measurable. Then the CQ channel E* is denoted by JCl

for any p E P (0).Namely, one of the QC channel is given in 2 9 . QC channel: Let {ITI,} be the set of all non negative Hermite operators with C , ITn = I, which is called a positive operator valued measure (POV). The QC channel Z* given by the measurement process using {n,} is obtained by x

n

(-1

for any u E 6 3-1 .

1.1. Noisy optical channel

To discuss the communication system using the laser signal mathematically, it is necessary to formulate the quantum communication theory being able to treat the quantum effects of signals and channels. In order to discuss influences of noise and loss in communication processes, one needs the following two systems 18. Let K l , K 2 be the separable Hilbert spaces for the noise and the loss systems, respectively. Quantum communication process is described by the following scheme 18

6 (3-11)

+

A*

+ 6 (3-12)

1

t

Y*

a*

1

t

6(?iiBKi)--+r*

--+

B(3-12BK2)

The quantum channel A* is given by the composition of three mappings a*, T*,y* such as

A* 3 a* o r * a* is a CP channel from 6 (3-12 @I

K2)

oy*.

to 6 (3-12) defined by

a* (a)= trlc2r 248

(6)

(7)

for any cr E B (3-12 @ Kz), where trKz is a partial trace with respect to K 2 . r* is the CP channel from 8 ('HI @ K1) to 6 ('Ha 8 I c 2 ) depending on the physical property of the device. y* is the CP channel from B (3-12) to B (3-11@ I c l ) with a certain noise state E E B (&) defined by (8)

7*(p)=LJ@t

for any p E by

B ('HI). The quantum channel A* with the noise < is written A* (PI = t r K z r* (P 8 <)

(9)

for any p E B ( ' H I ) . Here we briefly review noisy optical channel 27. A channel A* is called a noisy optical channel if 7r* and E above are given by

E = Im) (mi and r* (.) = V (-) V * ,

(10)

where Im) (ml is m photon number state in 'HI and V is a linear mapping from X1 @ K 1 to 'H2 8 K 2 given by n+m

v (in)8 Im))= C cj"ym l j ) 8 1n + m - j >,

(11)

j=O

Cj"'"

=

c K

r=L

(-1)"-

+

dn!m!j!( n m - j ) ! (ym-j+2r r! ( n- r ) !( j - r ) !(m- j r ) !

+

n+j-2r

(4

(12) for any In) in 3-11 and K 3 min{j,n), L = m a x ( j - m,0), where a and p are complex numbers satisfying laI2 IpI2 = 1 , and 7 = IaI2 is the transmission rate of the channel. In particular, p @ is given by the tensor products of two coherent states (0)(81 @ 16) (61, then r* ( p 8 E ) is obtained by

+

T*

( ~ 8=0p + p K ) (ae + p ~ 8j &,-I

+ G K } (-Be + S K ; J .

7r* is called a generalized beam splittings. It means that one beam comes and two beams appear after passing through r*. Here we remark that the attenuation channel At; l8 is derived from the noisy optical channel with m = 0. That is, the attenuation channel At; was formulated in1' on 1983 such as

(13)

Ag(P) = t r K c , K ; ( P @3 63) = trlczVo ( p @ 10) (00

249

G,

(14)

where 10) (01 is the vacuum state in B(Ic1),VOis the mapping from 7 d 1 8 l C 1 to 7-l~8 I c 2 given by

This attenuation channel is most important channel for discussing the optical communication processes. After that, Accardi and Ohya reformulated it by using liftings, which is the dual map of the transition expectation by mean of Accardi. It contains the concept of beam splittings, which is extended by Fichtner, F'reudenberg and Libsher concerning the mappings on generalized Fock spaces. The generalized beam splittings on symmetric Fock space was formulated by using the compound Hida-Malliavin derivative and the compound Skorohod integral. 2. Ohya S-mixing entropy and Ohya mutual entropy In order to discuss some physical phenomena, for instance, phase transitions, we had better start without Hilbert space, so that we need to formulate the entropy of a state in a C* system 2 0 J 2 J 5 . Let (A, G ( d ) , a ( R ) )be a C*-dynamical system and S be a weak* compact and convex subset of e(d).For instance, S = e(d); S = I ( a ) , the set of all invatiant states for a; S = K ( a ) ,the set of all KMS states. Every state 'p E S has a maximal measure p pseudosupported on exS (the set of all extreme points of S) such that $0 =

s,

(17)

WdP.

The measure p giving the above decomposition is not unique unless S is a Choquet simplex, so that we denote the set of all such measures by M,(S). Put

D, ( S )= { M , ( S ); 3 , u k c IR+and k

{pk}

c exS

(18)

k

where b((p) is the Dirac measure concentrated on an state (p, and define

H(p)

-

pk k

250

1% PIC

(20)

for a measure p E DIp(S).Then the entropy of a state defined by

'p

E S w.r.t. S is

which is called Ohya S-mxinig entropy. This entropy is an extension of von Neumann's entropy 16, and it depends on the set S chosen. Hence it represents the uncertainty of the state measured from the reference system S. When S = B(d), we simply denote S6(d)(p)by S('p) in the sequel. When A =B ('HI), the quantum entropy is denoted by S ( p ) = -trplogp

for any density operators p in 6 ( ' H I ) , which was introduced by von Neumann around 1932 16. It denotes the amount of information contained in the quantum state given by the density operator. The mutual entropy was first discussed by Shannon to study the information transmission in classical systems and its fully general quantum version was formulated in 'O. The classical mutual entropy is determied by an input state and a channel, so that we denote the quantum mutual entropy with respect to the input state p and the quantum channel A* by I ( p ; A*) . This quantum mutual entropy I ( p ; A*) should satisfy the following three conditions: (1) If the channel A* is identity map, then the quantum mutual entropy equals to the von Neumann entropy of the input state, that is,

I ( p ; id) = s( p ) . ( 2 ) If the system is classical, then the quantum mutual entropy equals to the classical mutual entropy. ( 3 ) The following fundamental inequalities are satisfied: 0

I I ( p ;A*) 5 s ( P ) .

In order to define such a quantum mutual entropy, we need the quantum relative entropy and the joint state, which is called Ohya compound state, describing the correlation between an input state p and the output state A*p through a channel A*. Ohya compound state D E (corresponding to joint state in CS) of p and R*p was introduced in l8l1', which is given by

251

where E derives from a Schatten decomposition 32 (i.e., one dimension orthogonal decomposition of p ) { p = EnXnEn} of p. Ohya mutual entropy with respect to the input state p and the quantum channel A* was defined in l8 such as

I (P; A*) = SUP s( w ,g o ) E

,

(23)

where 00= p 8 A * p and S (GE,g o ) is the quantum relative entropy defined by Umegaki 37,13 as follows: t r p (log p

- log G) (when ranp c EEF) (otherwise)

There were several trials to extend the relative entropy to more general This mutual quantum systems and apply it to some other fields entropy I (p; A*) satisfies all conditions (1)-(2) mentioned above, and it satisfies also condition (3) the Shannon's type inequality as follows: 0 5 I ( p , A * ) 5 min{S(p) , S ( A * p ) } . It represents the amount of information correctly transmitted from the input quantum state p to the output quantum system through the quantum channel A*. It is easily shown that we can take orthogonal decomposition instead of the Schatten-von Neumann decomposition 32. Ohya mutual entropy is completely quantum, namely, they describe the information transmission from a quantum input to a quantum output. When the input system is classical, the state p is a probability distribution and the Schatten-von Neumann decomposition is unique with delta measures 6, such that p = EnAndn. In this case we need to code the classical state p by a quantum state, whose process is a quantum coding described by a channel r* such that r*6, = crn (quantum state) and 0 E F * p = EnXngn. Then Ohya mutual entropy I ( p ; A*) becomes Holevo's one, that is, 2338322.

I (p;A* o I'*)= S (A*g) -

C

n

XnS (A*G,)

when En XnS (A*crn)is finite. Let (A, B(d), Q (G)) be a unital C*-system and S be a weak* compact convex subset of B(A). For an initial state 'p E S and a channel A* : 6 (d)4 B (B), two fundamental compound states are

=

1

(25)

'p 8 R*p.

(26)

w 8 A*wdp,

252

Gf is called Ohya compound state, which expresses the correlation between the input state cp and the output state A*cp. Then I * &*cp

= GpS

(27)

is a nondemolition lifting. Ohya mutual entropy w.r.t. S and p is

If (p;A*) L= S (a:, Go)

(28)

and Ohya mutual entropy w.r.t. S is defined as

Is (cp; A*) E sup { 1;

(9; A*) ; p E Mp ( S ) } .

When a state cp is expressed as cp = is given by

I (cp; A*) =

xk

pkcpk

(29)

(fixed), the mutual entropy

C phS(A*w,A*cp).

(30)

k

If A : B 4 A is a unitial completely positive mapping between the algebras A and B,that is, the dual A* is a channeling transformation from the state space of A into that of B,then

S(A*cpl, A*cpz) 5 S(cp1, c pd

(31)

Let A : B + A be completely positive unitial mapping and cp be a state of 23. So 'p is an initial state of the channel A*. The quantum mutual entropy is defined after l8 as

where the least upper bound is over all orthogonal extremal decompositions. Now we show two theorems proved in 26 according to the Holevo bound. Let us define an extension of the functional of the relative entropy. If A and B are two positive Hermitian operators (not necassarily the states, i.e. not necessarily with unit traces) then we set

S(A,B ) = trA (log A - log B ) .

(33)

There is the following Bogoliubov inequality.

Theorem 2.1.

26

O n e has

S ( A ,B ) 2 trA (log trA - log trB) 253

(34)

The following theorem gives us the bound of the mutual entropy I ( p ; A* o y*), where y* is a classical -quantum channel.

Theorem 2.2. 26 For a probability distribution p = { A , } and a quantum = 1, One has the following coded State U = y * p &Uk , 2 0, inequality for any quantum channel decomposed as A* = A; o A; such that A ~s u E p E i by a projection valued measure { E i } :

ck

xi

In the case that the channel A*, is trivial; A*,a = 0,the above inequality reduces to the bound obtained by Holevo ':

k

3. Comparison of various quantum mutual type entropies 3.0.1. Coherent Entropy and Lindblad entropy Let us discuss the entropy exchange For a state p, a quantum channel A* is defined by an operator valued measure { A j } such as 34914.

j

Then define a matrix W = (Wij)i,jwith 254

by which the entropy exchange is defined by

Se ( p , A*) = -trW log W. Using the above entropy exchange, two mutual type entropies are defined as below and they are applied to the study of quantum version of Shannon's coding theorem The first one is called the coherent entropy IC ( p ; A*) 35 and the second one is called the Lindblad entropy I L ( p ; A*) 6 , which are defined by 33910111@14131336.

IC ( P ; A*) E S (A*p) - s e I L ( P ; A*)

S (P)

+

A*) s(A*P)- Se ( P , A*) . (PI

1

By comparing these mutual entropies for quantum information communication processes, we have the following theorem 30:

Theorem 3.1. W h e n { A j } is a projection valued measure and dim Aj = 1 for arbitrary state p we have (1) 0 5 I ( p , A*) 5 min { S ( p ) , S (A*p)} ( O h y a mutual entropy), (2) IC ( p , A*) = 0 (coherent entropy), (3) I L ( p , A*) = S ( p ) (Lindblad entropy). From this theorem, Ohya mutual entropy I ( p , A*) only satisfies the inequality held in classical systems, so that only Ohya mutual entropy can be a candidate as quantum extension of the classical mutual entropy. Other two entropies can describe a sort of entanglement between input and output, such a correlation can be also described by quasi-mutual entropy, a slight generalization of I ( p , A*) ,discussed in ' O s 5 . 4. Quantum capacity

The capacity of purely quantum channel was studied in Let S be the set of all input states satisfying some physical conditions. Let us consider the ability of information transmition for the quantum channel A*. The answer of this question is the capacity of quantum channel A* for a certain set S ~6 (XI) defined by 23121126124,28,29.

Cf (A*) = sup { I ( p ; A*) ; p E S }

255

(39)

When S =6 (XI), the capacity of quantum channel A* is denoted by C, (A*). Then the following theorem for the attenuation channel was proved in 23. Theorem 4.1. 23 For a subset S, = { p E 6 (3-11) ;dims ( p ) = n } , the capacity of the noisy optical channel A* satisfies

C? (A*) = logn, where s ( p ) is the support projection of p. When the mean energy of the input state vectors { I d k ) } can be taken infinite, i.e.,

the above theorem tells that the quantum capacity for the noisy optical channel A* with respect to S, becomes logn. It is a natural result, however it is impossible to take the mean energy of input state vector infinite. Therefore we have to compute the quantum capacity

C,S. (A*) = SUP { I (pi A*) ;p E Se} (40) under some constraint Se = { p E S;E ( p ) < e} on the mean energy E ( p ) of the input state p. In we also considered the pseudo-quantum capacity Ct; (A*) defined by 20923,

cP"p.(A*) = SUP { l p (P; A*) ; P E se}

(41)

with the pseudo-mutual entropy Ipp( p ; A*)

c

Ipq ( p ; A*) = sup {

( A * p k , A*p)

;p =

c

xkpk,

k

k

finite decomposition

I

(42) where the supremum is taken over all finite decompositions instead of all orthogonal pure decompositions for purely quantum mutual entropy. A pseudo-quantum code is a probability distribution on B(7-L)with finite support in the set of product states. so { ( A h ) , ( p k ) } is a pseudo-quantum code if ( A , ) is a probability vector and p k are product states of B(3-1). The quantum states p k are sent over the quantum mechanical media, for example, optical fiber, and yield the output quantum states A * p k . The performance of coding and transmission is measured by the pseudo-mutual entropy (information) Ipq((Ak), (Pk);A*)

256

= Ipq

@;A*)

(43)

,

ck

with p = & P k . Taking the supremum over certain classes of pseudoquantum codes, we obtain various capacities of the channel. The supremum is over product states because we have mainly product (that is, memoryless) channels in our mind. Here we consider a subclass of pseudo-quantum codes. A quantum code is defined by the additional requirement that { p k } is a set of pairwise orthogonal pure states 18. However the pseudomutual entropy is not well-matched to the conditions explained in Sec.3, and it is difficult to compute numerically 24. From the monotonicity of the mutual entropy 22, we have

In order to estimate the quantum mutual entropy , we introduce the concept of divergence center. Let { w i : i E I } be a family of states and R > 0. We say that the state w is a divergence center for { w i : i E I } with radius 5 R if S(wi,w) I R

for every i E I .

In the following discussion about the geometry of relative entropy (or divergence as it is called in information theory) the ideas of can be recognized very well.

Lemma 4.1.

23

Let ((A,),

(Pk))

be a quantum code f o r the channel A* and 5 R for { A * p k } . Then

w a divergence center with radius

Ipr?((~k), ( P k L A*) 5 R. Lemma 4.2. 23 Let $0, $1 and w be states of B(K)such that the Hilbert space K is finite dimensional and set $Jx = (1- X)$O X$1 (0 5 X I 1). If S(&, w ) , S($J1,w ) are finite and

+

S($JX, w> 2

S($l,w ) (0 6 I 1)

then S($l,W)

+ S($O,$Jl)

I S($O,W).

Lemma 4.3. 23 Let {wi : i E I } be afinite set of states of B(K)such that the Hilbert space K is finite dimensional. Then the exact divergence center is unique and it is in the convex hull o n the states w i . Theorem 4.2. 23 Let A* : G(3-1) + 6(K)be a channel with finite dimensional K. Then the capacity Cp( A*) is the divergence radius of the range of A*. 257

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259

WHITE NOISE CALCULUS AND STOCASTIC CALCULUS

L. ACCARDI, ANDREAS BOUKAS*

AMS Mathematics Subject Classification 2000 : 60H40

1. Introduction The plan of this paper is the following. In section (2), after recalling some basic points of the structure theory of classical stochastic process, we illustrate how quantum probability can substantiate Hida’s vision about the ”elementality” of white noise, i.e. that any other (purely non deterministic) process can be in some sense “built” from it. In section (3) we briefly mention how the idea of white noise approach to stochastic calculus was suggested by the stochastic limit of quantum theory and how it lead to the idea of developing a calculus for the non linear powers of white noise. In section (4 this idea is illustrated in the case of the square of white noise). In section (5) we show how the classical processes, unified by the square of white noise emerged in different contexts of classical probability, mathematical physics, and statistics. In section ( 6 ) we describe our attempt, developed in past years, t o extend the results obtained for the square to higher powers of white noise. The obstructions which make this dream hard to achieve (the so called no go theorems) are quickly reviewed in section (7). Finally in the Appendix, following the original paper [Meix34], we give a proof of Meixner’s classification theorem. ‘centro vito volterra, universith di roma tor vergata, accardivolterra.mat.uniroma2.it web page: http://volterra.mat.uniroma2.it Work done within the european union research training network (rtn): “quantum probability - applications” web site: http://hyperwave.math-inf.unigreifswald.de/algebra/qpapplications

260

2. Elemental processes The structure theory of classical stochastic processes is one of the highlights of classical stochastic analysis and it has been accomplished through the work of generations of probabilists. Cornerstones in this long and broad avenue have been the Wald decomposition; the De Finetti, Kolmogorov, Levy, Khintchin structure theory of independent increment processes; P. Levy martingale representation theorem and his innovation equation

J X ( t ) = @ ( X ( S ) ;5 S t,Y(s),t,dt) which opened the way to Ito stochastic calculus and the corresponding theory of stochastic differential equations; the Hida-Cramer representation theory for Gaussian processes; the Kunita-Watanabe extension of Levy martingale representation theorem to non continuous trajectory martingales; the systematic work by Doob, Meyer and many others, . . . The intuitive picture emerging from these developments is that any "generic" stochastic process (i.e. whose trajectories are not too irregular) can be built starting from: (i) deterministic processes (ii) distribution derivatives of stationary independent increment processes {&> in the sense that it can be decomposed as a sum of integrals of such processes. Moreover, any stationary independent increment process { Zt >,in its turn, has the following LQvy-Ito decomposition :

Zt=mt+aBt+Xt where m is a constant Bt is a Brownian motion X t is a compound Poisson process Finally any compound Poisson process X t can be expressed as an integral of independent Poisson processes P,,t with intensity of jumps equal to U

and where the baricentric measure of this decomposition d P ( u ) is called the Levy measure and has support in R \ (0). These decompositionsjustify Hida's terminology according to which the i.e.r.v's (idealized elemental random variables ) are: 261

{&) the distribution derivatives of the standard Brownian motion

{Pu,t}the distribution

derivatives of the standard Poisson process with

intensity u and this fact can be symbolically expressed by the formula:

xt

= f ( t ,{ B ) ,

{Pu))= f ( t ,{&

, s E R ) ; {Pu,s , s E R,u E IW \ ( 0 ) ) )

(1) expressing (in a unique way) the stochastic process X t as a (nonrandom) functional of: (i) the standard deterministic processes: t H t (ii) the standard white noise: B, (iii) the standard Poisson processes with intensity u E R \ (0): Pu,s, Clearly equation (1) is purely symbolic because both 8,and Pu+are random variable valued distributions and there is no natural way to define a nonlinear function of a distribution. Notice moreover that, at this level of development, the standard white noise: B, and the standard Poisson densities P,,, appear on a totally equal level! In the mid 1970’s Hida initiated his programme on white noise analysis from which slowly it begun to emerge a more radical vision, namely: there is only one elemental process: standard white noise! I call this a ”vision” because it was never formulated as a precise mathematical conjecture, nevertheless the outstanding role of white noise is a constant in Hida’s mathematical thought [HidaOl]. This is indeed a bold vision, because apparently the noises B, and P,,, look like totally disjoint objects! The first substantial support to Hida’s vision came from Hudson and Parthasarathy discovery [HuPa84b] that, from the quantum probabilistic point of view, the standard Poisson process with intensity u can be considered as a ”function” of the quantum white noise. More precisely (cf. the end of this section for the definitions involved): Theorem: Let P,,t denote the classical scalar valued standard Poisson process with intensity u.Then

P(2L,t) = &LBt

+ &B,f + B$Bt -

This achievement was an important event for probability and created a great impression in classical probabilists such as P.A. Meyer, because it showed that the Brownian motion and the Poisson process which, in classical probability, are not apparently connected, in quantum probability become expressible as sums of three fundamental objects (two if one consid262

ers, as usual in physics, the pair {&, B,f} as a single object: the quantum Brownian motion) ’Notice however that the number process and the quantum Brownian motion are independent in the very strong sense that a sum of stochastic integrals over these processes is zero if and only if each integrand is zero. In other terms: the white noise point of view is necessary in order to achieve the full reductionistic programme (Hida’s vision) of expressing the two building blocks of classical stochastic processes (Wiener and Poisson) as functions of a single more fundamental object: the quantum white noise. Notice in addition that, in order to achieve this reduction one needs the ”square modulus of the white noise” B$Bt = (which is for the number process what the white noise is for Brownian motion). This ”quadratic functional of the white noise” is the first example of the basic role played by the ”nonlinear powers the white noise”. Using the above reduction Parthasarathy [Parth92] proved that any infinitely divisible distribution can be represented as vacuum distribution of a stochastic process in the space of the Boson Fock Brownian motion. This result has now been developed into one of the basic new ideas of quantum probability: the quantum decomposition of a classical random variable [AcBo97]. Another important step in the direction of Hida’s vision will be discussed in the following section.

3. White noise approach to classical and quantum stochastic calculus The white noise approach to classical and quantum stochastic calculus came out from the experience accumulated in the solution of many concrete problems in physics and in mathematics. The main steps in this development are: 1) The development of quantum stochastic calculus: Hudson and Parthasarathy (1982) 2) The proof that quantum stochastic differential equations are ”fast time” (t 4 t/A2) limits of Hamiltonian equations: Accardi, Frigerio, Lu (1987) 3) The proof that white noise Hamiltonian equations are ”fast time” limits of Hamiltonian equations: Accardi, Lu, Volovich (1993) 4) The combination of 2) and 3) into the statement that (classical and quantum) stochastic equations can be expressed as (causally) normally or263

dered forms of (classical and quantum) white noise Hamiltonian equations [AcLuVo99], IAcLuVo021. The equivalence between white noise Hamiltonian equations and stochastic differential equations is quite nontrivial and the connection between the coefficientsof the two types of equations is strongly nonlinear. The advantage of the white noise Hamiltonian equations over their (classical or quantum) stochastic equivalent is that the formal unitarity conditions are the obvious ones, used by every physicist, which correspond to the formal self-adjointness of the Hamiltonian. The relationship expressing the coefficients of the stochastic equation as (nonlinear) functions of the coefficients of the corresponding white noise Hamiltonian equation explains the deep origins of the Hudson-Parthasarathy unitarity conditions and gives a "microscopic" interpretation of them in terms of the original (non stochastic neither white noise) Hamiltonian equations of which the stochastic and white noise equations are approximations. A systematic development with several new results can be found in the PhD thesis of Wided Ayed [AyedOS]. Recall that the standard scalar valued, classical white noise is a classical mean zero, Gaussian, operator valued distribution process with variance (W,Wt)= 6 ( t - s )

.

This implies that the standard white noise is a stationary, additive, independent increment process. Definition: A quantum stochastic process is a family z=

{ z ( t ) : t E Rd}

of Hilbert space operators. Such a process is said to be classical if (i) For all t 2 0, each z ( t ) is a QM "observable" i.e

z(t) = z(t)*

;

w

(ii) For all t , s,

[z(t), z(s)] = z ( t ) z ( s ) - z ( s ) z ( t ) = 0 the commutator (here and in the following) will always be meant weakly on some dense domain. Definition: A boson Fock (&dimensional) white noise (equivalently called, in the physical literature, a free boson Fock field over L 2 ( R d ) )is a pair bt, b$ of operator valued distributions acting on a Hilbert space 3-1, satisfying the commutation relations:

[bt,b:] = 6 ( t - s)

t

E

264

Rd

[algebra]

and such that there exists a unit vector 6, E ‘FI satisfying: [Fock prescription].

bt6, = 0

For the precise meaning of these expressions, including domains, etc. ... cf. [AcAyOuOS]. The definition above best illustrates one of the basic general principles of QP namely that: algebra implies statistics. In fact the above definition of white noise, which is purely algebraic, implies the Brownian motion statistics in the sense that it implies that the random variables

Wt-W, :=

I’

dU(b:+b,)

Wo = O

;

mutually commute, are independent on disjoint intervals and (a,eiWt+j = ( a , , i S , t d s ( b , f + b s ) + ) = e - 7t 2 equivalently:

Wt = Bt

+ B$ =

I’

ds(bz

+ b,)

is the increment process of a classical Brownian motion (this is the quantum decomposition of the classical Brownian motion). Similarly one has the quantum decomposition of the classical white noise Wt

= bt

+ b,f

In this sense one can say that in classical probability wt is ”elemental” (atomic), but in quantum probability it is not. The connection between (the increment process of) quantum Brownian motion and quantum white noise is the same as in the classical case:

B$

=

I

t

dsb:

, Bt=

I

t

dsb,.

Conversely a naive approach would suggest the conjecture that the relation between quantum Brownian motion and quantum white noise is:

d , btf := -B$. d bt := -Bt dt dt However application of this rule without constraints leads to the wrong results as illustrated by the Hamiltonian white noise equations. 265

4.

Nonlinear powers of white noise

Given the identification between the standard (d-dimensional) quantum white noise and the free boson Fock fields over L2(Rd),one can formulate what has been one of the fundamental unsolved problems of theoretical physics since over 30 years: to give a reasonable construction of ”local powers of free quantum fields”. This means that one would like to associate well defined mathematical objects to symbolic expressions of the form:

in such a way to preserve as many properties as possible of the discrete approximations of these objects (which are well defined!). In other terms we would like to give a meaning to the ill defined ”powers of an operator valued distributions”. Since these objects are very singular some ”renormalization” is needed to achieve this goal. In the literature one can find many ”renormalization techniques”. The usual technique consists in introducing appropriate cut-offs and then trying to remove them with some limiting procedure and after having subtract some quantities tending to m. The main problem met up to now with this technique is the following dichotomy. After renormalization the resulting field is: (i) either trivial (Gaussian or simple perturbation thereof) (ii) or completely uncontrollable This dichotomy is well illustrated in the paper by Segal [Sega’lOa] which is devoted to the simplest renormalization problem: that involving the squares of quantum fields. Segal proves that the square of the usual time zero, scalar, Fock, KleinGordon field on E d , cannot correspond to a self-adjoint operator acting on the same Fock space of the field unless d = 1. However his techniques are not able to provide: - any information where this operator lives - any information on its spectral distribution in interesting states Thus, even for squares of fields the situation was very obscure until 1999. In 1999 a new idea was introduced in this problem. The main point of this new idea was to postulate new, renormalized, commutation relations for the higher powers of white noise and then construct some Hilbert space realization of them. More precisely: one starts from the standard boson 266

algebra

[bt,b:] = 6 ( t - S ) and computes formally the higher powers commutators

[b!, b:kl combining the 1-st order commutation relations with the renormalization rule:

a(t - s

- s)

) = ~ cn-16(t

VnE

N \ (01

Then one takes the result of this manipulation as the definition of the Lie algebra of the renormalized higher powers of white noise. The second much more difficult step consists in trying to construct some Hilbert space realization, for example by introducing the Fock prescription:

h > l

;

b;‘“b:@=O

which, also in this case, uniquely determines the statistics. For the second powers, in the Fock case, this programme was realized in [AcLuVo99] and led to the definition of the renormalized square of white noise (RSWN) as the Lie algebra with generators

b$

lL=”

/dtcp(t)b: nrp lL=”

/

;

b, = (b$)+

dtcp(t)b$bt

where cp is some test function, and Lie brackets given by the commutation relations:

[bb$I

= Y(cp7 $)

b,

+ nq*

b;ZI = 2 q *

(b$)+ = b ,

;

n: = nv

where y is a strictly positive constant called ”the renormalization constant”. The Fock representation of this Lie algebra consists in realizing these operators on a Hilbert space ‘H with a unit vector @ satisfying:

b,@=O. 267

Theorem: (Accardi, Lu, Volovich 1999) The Fock representation of the renormalized square of white noise (RSWN) exists. Soon after this result it was recognized by Accardi, Franz and Skeide [AcFrSkOO] that the Lie algebra of the renormalized square of white noise is isomorphic to (a central extension of) sZ(2, R). This has 3 generators

B+,

B-,

M

and relations

[B-,B+]= M

[ M ,B*] = f 2 B * . The paper [AcFrSkOO],combining the well developed theory of unitary representations of sZ(2,R) with Schiirmann’s representation theorem for quantum independent increment processes [Schii93], obtains: (i) a classification of all the unitary representations of this current algebra satisfying an irreducibility condition (ii) a classification of all the classical sub-processes of the RSWN as well as the identification of their vacuum distributions. An unexpected fall out of this identification was an unexpected connection with the Meixner classes [Meix34]. In order to illustrate this connection let us first consider the case of the 1-st order quantum white noise b$, bt and the family of classical (selfadjoint) processes that can be obtained by linear combinations from it and the numbers process bzbt. A simple calculation shows that these processes are of the form

z ( t )= a

+ z B ( t )+

+ PN(t)

where a,p E R and z E C. In more usual white noise notations: k ( t ) = Q!

+ z bt + .Zb$ + ,f3b$bt .

The constant part a is central and we omit it. Also the case z = 0 is trivial and we omit it. Then, up to the ”time rescaling” t H t/lzl, the ”gauge transformation” bt H eiebt, where z = lzleie and a renaming of p, we are left with the family k ( t ) = bt

+ b z + ,BN(t) .

Remembering the Hudson-Parthasarathy decomposition of the white and Poisson noises, we see that, depending on the parameter p, this family only 268

1 critical case, namely: p = 0 which corresponds to classical scalar valued standard Brownian motion

x ( t ) = b(t)

+ b+(t).

If /3 # 0 then, fixing a complex square root of

A=+ after another rescaling and gauge transformation we obtain:

+

+

z(t)= JS;B(t) A B + ( t ) N ( t ) which is the classical scalar valued standard Poisson process with intensity A. A similar calculation applied to the RSWN shows that the family of classical processes that can be obtained by linear combinations from the 2-d order processes b z 2 , bf and the numbers process b$bt has the form:

z ( t )= a

+ zb,2 + Eb,+2 + pb,+bt

where a , P E R and z E C. Again up to centering, rescaling and a gauge transformation: x p ( t ) := bZ2

where cases:

+ b," + p b z b , = B$ + Bt + PNt

p is a real number. It can be proved that here there are 2 critical p=f2

the value +2 corresponding to the square position (classical) white noise, i.e.

I b,++bt 12=

b,+2+b,2+b,+bt+btbt++ = by+b,2+2b;bt+s(o)

3 b,+2+b,2+2b,+bt

and the value -2 to the renormalized square of the momentum white noise, i.e.

(b,+ - b t ) / i the vacuum distribution of both these critical processes is the Gammadistribution

whose parameter mo > 0 is uniquely determined by the choice of the unitary representation of sZ(2, R) corresponding to the representation of the renormalized square of white noise algebra [ACFRSKOO]. 269

In this functional realization the number vectors become the Laguerre polynomials which are orthogonal for the gamma distribution. Since the Gamma-distributions are precisely the distributions of the X2-random variables, this result confirms the naive intuition that the distribution of the renormalized square of white noise should be a Gammadistributions For IpI < 2 the jumps are not strong enough and one still has a density

where C is a normalization constant. The orthogonal polynomials corresponding to this probability measure are the of the second kind, or MeixnerPollaczek polynomials. For mo integer or half-odd there are explicit formulae for the densities due to Grigelionis:

. . ( ( n- 1 ) 2 + x2) Ir ( n+ ix)12 = 7rx(1+ x2).sinh(nx) , 2

lr(i+ix)l 1

.(a

IT

= cosh(7rx)

+ 2). ..((n

, X€R

, x€R

+ a + z2)

- 1)n cosh(~ x )

n = 1 , 2 , ...,

,

n = 1 , 2 )...; 2 E R .

> 2 the jumps dominate and the probability measure is Finally, for atomic, namely the negative binomial (Pascal) distribution:

where ( r n ~ denotes )~ the Pochammer symbol (mo)n= mo(m0

+ 1) . . . (mo+ n - 1)

and

270

ifp>+2

< -2. The orthogonal polynomials associated to a centered Pascal (negative binomial) distribution are the Meixner polynomials of the first kind:

if /3

if /3

> +2 and

ir

n+mo-l

k=l

5. Emergence of the square of white noise in different

contexts The connection between the gamma processes and the current representations of SL(2,R) was studied in [TsiVeYoOl] independently of [AcFrSkOO] (where this connection was established for all the Meixner classes). The Lkvy processes, corresponding to the Pascal measures were introduced by Bruss and Rogers [BruRoSl] in the context of optimal selection strategies based on relative ranks, when the total number of options is unknown. In the paper [GrigOl]Grigelionis uses the term “Meixner distribution” to denote the class of probability measures on R whose characteristic function (Fourier transform of the probability density) has the form

with z E R,-7r < p < 7r, S > 0 , p E R. This class of probability measures was introduced by Schoutens and Teugels [SchTeu98]who established their connection with the Meixner-Pollaczek polynomials and proved that the measures in this class correspond to Levy processes (the explicit construction of the Fock representation in [AcLuVoSg]can be considered a different proof of this result). The papers by Nualart and Schoutens [NUSCHOO] and by Schoutens and Teugels [SchTeu98]study the gamma, Pascal, and Meixner processes as main examples of generalized chaotic representation for squareintegrable 271

random variables in terms of the orthogonalized Teugels martingales (which are the centered power jump processes related to the original process). They use the onedimensional polynomials of Meixner’s type in order to carry out the orthogonalization procedure of the Teugels martingales . We refer to [GrigOl], [Grig99], [GrigOOc]for several interesting properties of these distributions and explicit formulae related to them. In particular, in [GrigOl], the Meixner process was proposed as a model for risky assets and an analogue of the Black and Sholes formula was established for them. The infinite dimensional and multidimensional analogues of orthogonal polynomials associated to a given measure have been widely studied both in the Gaussian case and in the Poisson case (CHIHA, KOKUOL). The programme to extend this analysis to more general probability measures was developed by Berezansky [BEREZb98], [BEREZa97], [BEK094], [BELILY951 who introduced in this connection the notion of Jacobi field of operators, and his school [LYTVO2a], [LYTVOSb],[LYTV95c]. An infinite-dimensional analogue of the Laguerre polynomials and the associated Jacobi fields, corresponding to the gamma case, i.e. to the class (111) in Meixner’s classification, was studied in [KonLitOO], [KoSiStr97]. In conclusion it should be added that the square of white noise (RSWN) was introduced as an example of interacting Fock space and in the attempt to extend to infinite dimensions the canonical connection between orthogonal polynomials and interacting Fock spaces established, in the 1-dimensional case, in [AcBo97] and, in the 1-dimensional case, in [AcKuSt02]. I

6. Higher powers of white noise The next step of our programme is: to extend, if possible, the results obtained for the square to higher powers of white noise. We are developing this programme jointly with Andreas Boukas including collaborations with Uwe Franz, in the attempt to overcome the obstructions posed by the no-go theorems (cf. below), and with Rene Schott and Massimo Regoli, on Lie algebra and algorithmic aspects: among other things we are trying to extend the symbolic programme developed by Feinsilver and Schott [FeinScho93] for calculations on Lie algebras. Our programme is to characterize those independent increment stationary processes (SIIP or Levy processes) which arise as renormalized higher powers of the standard (Fock) quantum white noise. For a short period, in 272

2003, we believed we had realized this dream but, as I will try to explain below, the situation is more subtle and the problem is related to some long standing open problems in the classical theory of SIIP. Definition The renormalized boson Fock white noise ( simply RBF white noise in the following ) over a Hilbert space 'FI with vacuum (unit) vector @ is the locally finite Lie *-algebra canonically associated to the associative unital *-algebra of operator-valued distributions on 'FI with generators b;",

tERd

k, EN,

bt,

and relations [bt,bf] = 6 ( t - s)

p,t,b 9 = lbt , bsl = 0 (b,)* = bi bt@=O

6(t)' = c 1 - ' 6 ( t )

,

c > 0,

1 = 2,3, ....

Lemma The Lie algebra, associated to the RBF white noise (renormalized boson Fock white noise), is the Lie algebra with generators b,tkb2" =: bE(t) central element b:b,t" =: E and relations

(bf"2")t

= (bt")b!

where: k = 0,1,2, ...,

k(l) =

k(k - l)(k - 2) . . . (k - I + 1)

k(') = o if I > IC

These conditions guarantee that no negative powers of the white noise functionals appear. 273

In terms of the smeared generators

with involution

(B,"(fN*=

mf)

and central elements

the relations become

Now let us deduce some necessary conditions for the existence of the Fock representation. Lemma (Boson Independent increments) Suppose that in the scalar product

the supports of any two test functions either coincide or are disjoint. Denote by Z the family of all supports of all the test functions appearing in the scalar product. Then the above scalar product is equal to

where if {A : s u p p ( 4 ~ )= I } = 8 we interpret

as 1. 274

7. No go theorems The main result of [AcLuVo99]was the existence of the Fock representation for the second order white noise. In [AcFrSkOO] it was shown, among other things, that this representation can be interpreted as a representation of the current algebra over the Lie algebra sL(2, W). The analogue representation for the first order white noise, which corresponds to the Heisenberg-Weyl algebra, had been known in physics for over 70 years. Now: a current algebra over a Lie algebra is a functional version of the Lie algebra itself. More precisely it is an algebra of functions on some measure space ( X ,v), (which in the case of [AcLuVo99]was E X d ) with values in this Lie algebra (such algebras were introduced and widely studied in the 1960's and, in the more recent mathematical literature, they are sometimes called " Kac-Moody algebras"). Now it might seem, at first glance, natural to conjecture that, if a Lie algebra has a Fock (lowest weight) representation, then the associated current algebra too have one. This is certainly true if the measure space ( X ,v) has a finite number of points because in this case the current representation is a finite tensor product of the original one. For example the Lie algebra generated by the Heisenberg-Weyl algebra and sL(2,R),called the Schrodinger algebra, has been widely studied in the literature and, since the Schrodinger representation exists in any finite dimension, the associated current algebra over ( X ,v ) has a Fock representation for any space X with a finite number of points. In the paper [Snia99], devoted to the extension of the results of [AcLuVo99] to the free case, Sniady proved the following result. Theorem The joint Fock representation of the first and second order white noise, i.e. of the Schrodinger algebra, cannot exist. This theorem was generalized in [AcFrSkOO] and further generalized by Accardi, Boukas and Franz [AcBouFk05] whose result, reported below, destroyed the hopes of a naive generalization, the higher powers of white noise, of the results obtained for the second power.

Theorem 7.1. In the notation (2), denote

where X I is the characteristic function of the interval I & W (taking value 1 on I and 0 elsewhere). Let C be a Lie *-algebra with the following properties: 275

(i) L contains B,", and Bin (ii) the BE satisfy the higher power commutation relations . Then L does not have a Fock representation i f the interval I is such that

where c denotes the renormalization constant. This theorem means that we cannot hope to have a single representation including all the higher powers of white noise: the best one can hope is to form, for each n,the smallest Lie algebra generated by B," and B: and look for a representation of it. The difficulty with this programme is that, as soon as n 2 3 these Lie algebras are infinite dimensional and not so widely studied. In particular one cannot apply the general methods of [AcFrSkeOO], which heavily used the known theory of irreducible unitary representations of s l ( 2 ,R), and one has to go back to the direct method of [AcLuVoSg]which however, in these cases is much more complex due to the more complex structure of the higher order commutation relations. At the moment we do not know if such a representation exists even in the case n = 3. The following considerations show that this difficulty is related to and old open problem of classical probability. In the case of 2-d order noise and of higher orders with a single renormalization constant, the current algebra restricted to a single block Liespan { B , h ( ~ ~ is ~ ,isomorphic ~l)} to the 1-mode Lie algebra Lie-span-{a+'ak}

Lemma 7.1. Let (b:) be the Boson Fock scalar white noise. Suppose that the k-th power of white noise exists for some natural znteger k and admit a Pock representation. then the process

{wi,t],a,-OO
+00}

defined formally by some renormalization of

should be a stationary additive independent increment process o n R. 276

Lemma 7.2. The map

with the [AcBouFrO5] renormalization, is a Lie algebra isomorphism. Corollary. The Fock statistics of (B::),])is the same as that of (uk).

Proof. This statistics is uniquely determined by the Lie algebra structure.

Lemma 7.3. The vacuum distribution of

1

1

(bt

+ bt)"dt =

1

1

wpdt

coincides with that of

+ a)n

(u+

Proof. The statistics is uniquely determined by the Lie algebra structure Corollary. If the Fock representation of the n-th power of white noise exists, then the vacuum distribution of

(a+

+ a)n

must be infinitely divisible.

+

Proof. From Lemma (7.3) it follows that the distribution of (a+ a)n is the same as the distribution of dt(b? + bt)" and from Lemma (7.1) we know that this is infinitely divisible.

Ji

Theorem 7.2. A necessary condition for the existence of the n-th power of white noise, renormalized as i n [AcBouFrOS] is that the n-th power of a classical Gaussian random variable is infinitely divisible. In classical probability the n-th powers of the standard (1-dimensional, mean 0, variance 1) Gaussian random variable and their distributions have been widely studied. In particular it is known that, Vk _> 1,y Z k is infinitely divisible. However it is not known if, Vk 1 1, yZk+' is infinitely divisible. This suggests the conjecture that the above mentioned programme might be realizable if one starts from even powers (which fortunately are closed under Lie brackets). 277

Appendix: Meixner’s classification theorem 8. Orthogonal generating functions

The purpose of this appendix is to give an exposition of the problem studied by Meixner and of its method of solution. To this goal we begin with some general definitions.

Definition 8.1. Let p be a probability measure on R with moments of any order and let Pn(x) (x E R;n E N), denote the orthogonal polynomials of p normalized so that

&(x)

= 1;

leading term of P,(z)

=

1.

(4)

A function

is called an orthogonal generating function if

where the series in (6) converges weakly in L2(R,p).

Problem. Under which conditions is a function F : R x R the orthogonal generating function of some probability measure on R? It is easy to verify that a necessary condition is that, denoting (., .) the scalar product in L2(R,p), one has ( 1 , J Y . , t ) )= (Po,F(.,t))= 1 * In other words

Theorem 8.1. Now suppose that condition (7) is satisfied with a function F ( x , t ) of the special form ~ ( zt ), = e z u ( t ) f ( t )

(8)

where u : IR -+ R is an invertible function such that u(0)= 0 278

(9)

and f is a function such that

(11)

u’(0)= 1 .

If one assumes that u is invertible, then the Laplace transform of p is uniquely determined in its domain, by the formula I

P

Proof. Then (7) becomes

f ( t ) p ( d z )= 1

.

Introducing the change of variable U(t)

t = U-’(‘T)

=: 7 ;

the identity (15) or equivalently

becomes (12).

Remark. The meaning of Theorem (8.1) is that a probability measure p, satisfying (7) and (8), is uniquely determined by the pair ( f l u ) provided that u is invertible. In his paper [Meix34]Meixner: (i) determines all pairs of functions (f,u ) satisfying conditions (7) and (8) for some probability measure p (ii) shows that for each such pair (f,u),u is invertible (iii) explicitly determines all the corresponding probability measure. This justifies the following

Definition 8.2. A probability measure p on R is called a Meixner measure if (i) p admits an orthogonal generating function F ( z ,t ) (ii) F ( z , t ) has the form (8) for some pair of functions ( f , u )called the associated pair. 279

Finally let us prove that the ansatz (8) is coherent, i.e. that the series expansion of its right hand side has the form (6) with the P,(x) satisfying condition (4).This follows from the following:

Lemma 8.1. Let f ( t ) be a formal power series with constant term f(0) = 1

and let u(t) be a formal power series with constant term u ( 0 ) = 0 and with linear term coefficient u1 = 1, i.e. u ( t )= t ( l

+ [t]+ . . . )

Then there exist polynomials P,(x),with leading coefficient equal to 1,

Pu(x)= xn + an,lzn-'

+ . . . + an,,

such that the following formal expansion holds

Proof. By assumption

u ( t )=

c

=t

U,t"

+ ii2(t) ;

uo = 0 ;

u1 =

1

.

"20

We define the degree of a formal power series Cn,Oanzn, the smallest n E N such that a , # 0. For example, f has degree 5 , u has degree 1 and i i 2 degree 2 2. Moreover

Our assumption on u implies that

+ deg iiz,"(t) 2 n + 1

u(t)" = [t" ii2"(t)] (u2")

and

f (9 = [I+ with deg f ( t ) 2 1. Therefore

u(t)"f ( t )= [t+iiz(t)]"[l+f(t)] = [tn+ii2,,(t)l[1+f(t)] = tn+G2,n+t"f+62,,f 280

with deg tnf

2 n + 1 ; deg i i 2 , n f _> 2(n+ 1)

therefore

u ( t ) " f ( t )= tn with deg B,(t)

+ Bn(t)

2 n + 1. Therefore

with deg Cn+l(t)2 n

+ 1. Therefore

which proves that the leading coefficient of Pn(x) is equal to 1. 9. The equations for f and for

21

= 21-l

Denote v the inverse formal power series of u,i.e. by definition

u ( v ( t ) )= v ( u ( t ) )= t and denote

d D := dx Then the following identity is clearly satisfied:

v( D)e""(t)f(t ) = v(u(t))e2"(t) f ( t )= te""@)f(t) Taking

a;

(17)

of both sides of (17) one finds

+

v (D)a;ezu(t)f ( t ) = a: (tezu(t) f ( t ) = ta,"eTu(t)f ( t ) na,"-l ezU@) f(t)

evaluating this a t t = 0 and keeping (17) into account, one finds

v(D)Pn(x)= nPn-l(x). (18) On the other hand the Pn(x) are the orthogonal polynomials of some measure II, on R if and only if there exist two sequences (In) and ( k n ) of real numbers such that knIO; and

281

b'n

with the convection that P-1(2) = 0

Denoting x the multiplication by x and using the identity

[V(D), I. = W

)

we find, combining (18) and (19): v(D)Pn+1(z)= ( n + l ) P n ( ~= ) (z+ln+l)nPn-l(z)+v’(D)Pn(2)+Icn+l (n-l)Pn-z(z)

(20) while the usual Jacobi relation (19) is

n P n ( x ) = (x

+ ln)nPn-l(x) + nknpn-z

.

(21)

Subtracting (21) from (20) one finds

or equivalently

Applying v ( D ) to both sides and dividing by n

+ 1 one obtains

Now, P,(x)cannot be identically zero because its leading coefficient is equal to 1. Therefore comparing (22) and (23) we conclude that there exist constants A, K such that ln+l - 1,

kn+1 n

=

kn = K n-1

x * ln+l

= nx

+1

* kn+l = n((n- 1). + k z ) .

Notice that, since the k, are all negative, also Moreover, given (24) and (25), (19) becomes:

Pn+l(z)= (x + 11

K

must be negative.

+ nX)Pn(x)+ n ( h + ( n- l ) K ) P n - l ( X ) . 282

(24) (25)

We know that k2 5 0 and it cannot be = 0, otherwise $ is a multiple of a &measure, hence in (16) Pn(x) = 0 , V n > 1 and, since PO(,) = 1 by assumption, equation (16) becomes

f ( t )= e - Z U @ ) which can be satisfied for any x , t E R if and only if

u ( t )E 0

, f ( t )= 1

which corresponds to a trivial solution. Thus for all non trivial solutions one must have

Since

it follows that

C nXPn(0)-n! = At c~ ~ ( 0( n) -- l)! = Xtf’(t) tn

tn-1

n20

C n>l

n>l

tn k2pn-1(0)-

( n - I)!

tn-1

= kzt

C Pn-l(0)-( n- I)!

n2l

= k2t

c

tn Pn(0)-

n>O

= Kt2f’(t)

From these identities one deduces that

+

+

+

f ’ ( t )= Zif(t) Mf’(t) k z t f ( t ) Kt2f’(t) 283

n!

or equivalently f satisfies the equation

Moreover from (18) and (22) we find

+ KW(D)~P,(~) ; Vn

(1- v’(D))P,(z) = Av(D)P,(s)

(27)

Therefore, as operators on Liol(R, p )

1 - v’(D) = Av(D)

+KV(D)2

or equivalently

v’ = 1 - Av - nu2

.

Therefore the pair (f,v) (equivalently (f,u ) ) is uniquely determined by the solutions of the equations (26), (28) respectively. Notice that ‘the same polynomial

1- At - K t 2

(29) appears in both equations. According to the various possible values of the parameters A, K , we distinguish 5 possibilities: (I) A = K = 0 ((29) has degree 0) (11) K = 0; A # 0 ((29) has degree 1) (111) A2 = - 4 # ~ 0 (29) has degree 2 and one non zero root of multiplicity 2) (IV) A2 > - 4 ~> 0 (29) has degree 2 and 2 distinct non zero real roots) (V) 0 < A2 < -46 (29) has degree 2 and 2 non zero complex conjugate roots) The five Meixner classes are defined by t the solutions of equations (as), (26) corresponding to the values of the parameters ( A , K ) in the classes defined above.

Remark. In fact Meixner ([Meix34], Section 6 ) calls class (11) what we have called class (111) and conversely. Moreover Meixner does not classify his five classes in terms of the parameters (A, K ) but in terms of two auxiliary parameters ( a ,p), related to (A, K ) by the equations a+P=A

ap = - K . 284

In the following section we will describe the translation code between our parametrization and Meixner’s. 10. Meixner’s parametrization

Theorem 10.1.

(i) For any real numbers X,K there exist complex numbers that the following identity holds

a,P such

1 - At - K t 2 = (1- at)(l- p t )

(30)

a+p=x

(31)

ap = -K

(32)

or equivalently

(ii) The pair (A,&) uniquely determines the pair ( a l p )up to the permutation

(a,P) (iii)

+

(P,a )

If in addition K I O

(33)

then there are only four possibilities

K

X=K=O*a=P=O

(34)

x#0H

(35)

=0 ;

( a ,p) = (A, 0)

(37)

where, in all the above identities ( a ,P ) has been identified to (p,a ) and the square roots are the positive ones. Moreover the last possibility (37) splits into three according to the following situations:

285

i.e. only one real solution

i. e. two distinct real solutions

i.e. two complex conjugate solutions. (iv) The five Meixner classes are characterized by the following values of the pair (A, 6): (I) x = r; = 0 (11) A, n # 0, x = f 2 1 4 1 / 2 (111) n = 0; x # 0 (IV) A, r; # 0; A2 > 41nI (V) A, n # 0; A2 < 4)nI. Let us first discuss the equation

0 = 1- A t - r;t2

If X=rC=O

there are no solutions.

If n=o;

A#O

(43)

there is only one solution

1

t,, = -

x

*

If n#O;

x=o

then (41) becomes

+

0 = 1 - nt2 = 1 Inlt2 which has only 2 purely imaginary complex conjugate solutions:

t&f =f i 286

1:/

-*

(44)

If both

then equation (41) can be written

x

1

O=(K.lt2-Xt+l~O=t2--t+IKI

14

=

(

t--

1

--

21 ;)2

4K12

+m

which has exactly one solution if and only if (38) holds. In this case the solution is tl,

=

x

214

and, due to the relation (38) there are 2 possibilities (46)

giving rise to the solutions 1

t1%* =

*/K11/2

(47)

The two remaining possibilities, beyond (38) are (39) and (40). Condition (39) corresponds to two distinct real solutions

Condition (40) corresponds to two complex conjugate solutions

Now let us consider the identity (30) which is equivalent to 1- At - K t 2 = 1- ( a

+ p>t+ apt2

It is clear that the pair ( a ,p) is a solution if and only if the pair Equating coefficients we find (31), (32). Fkom these we deduce -6

= (A - p)p = xp - p2

i.e.

287

(p,a ) is.

This gives the solutions

which satisfy the condition

(P+,"+)

= ("-1P-1.

Let us discuss the possible solutions of the system (32), (31) corresponding to the various possibilities for the parameters X and K . (34) is obvious. Clearly (42) holds if and only if

a=p=o. Now suppose that (44) holds then

p-=o

P+=X#O;

a+=o; a-=X#O that is, exactly one number, in the pair (alp)is # 0. Conversely, if this is the case, then X must be # 0, otherwise

a* = -P* and it is impossible that exactly one is be fulfilled only if X2/4 IC is real. In this case one has always

+

p+>o;

# 0.

Moreover this condition can

"->O.

Thus the condition that exactly one in the pair ( a ,P ) is different from zero can be fulfilled only if either 2

-+K=o

or

Thus the two conditions coincide and are both equivalent to K=O.

288

This proves (35). Now suppose that condition (44) holds. Then the system (32), (31) becomes

a=-@ K = p

2

(51) (52)

Since K # 0, this means that @ must be purely imaginary and # 0. Conversely, if this is the case and (51) holds, then (45) holds. This proves (36).

If condition (45) holds, then the system (32), (31) has 2 distinct solutions satisfying

(a+,P+),(a-,P-)

= (P+,a+)

(53)

Conversely, if this is the case, then (45) must hold because, if either X or K are zero, then (53) cannot define two distinct solutions. Finally note that the above discussion is valid in both cases when the solutions of (41) are real or complex, i.e. if either condition (39) or (40) hold. This proves (37). This completes the proof of (iii). The 1-st Meixner class is clearly characterized by the condition X=n=O

The condition characterizing the 2-nd Meixner class is equivalent to the case (37) under condition (38), i.e. when equation (41) has a unique non zero real solution. The 3-rd Meixner class is equivalent to the case (35). The 4-th Meixner class is equivalent to the case (37) under the condition (39), corresponding to two distinct real nonzero solutions. The 5-th Meixner class is equivalent to the case (37) under the condition (40), corresponding to two complex conjugate solutions.

11. Solutions of the equation for

21

In the present section we discuss the solutions of equation (28) corresponding to the various Meixner classes. Class I: A = n = 0. In this case equation (28) becomes v' = 1 289

(54)

By assumption u ( t )= t

+ tZuz(t)

where uz(t) is an arbitrary formal power series. Moreover

Thus

0 = v(0) = 210

(55)

and the unique solution of (54) with initial condition (55) is v(7) = 7- .

(56)

Class 11: A, tc # 0 , X = f 2 1 ~ 1 ' / ~In. this case equation (28) becomes w'(7)

= 1 2)K11/2v(7-)

+

v(T)21K.I

= (1

This gives

which is of the form

with b=O;

a=l;

c = l t p *

Therefore the solution is

This gives

or 1

(t

+

C)lK[

1 = v(t) lKp2

and condition (55) is satisfied if and only if

290

lK1'/221(7-))2.

This gives

In this case equation (28) becomes

Thus

1

t + c = -- ln(1- Xu)

x

2,

=1 (1 - e-x'e-xc)

,

X The condition

v(0) = 0 fixes c = 0, so that

1

= - (1- e - X T ),

x

Class IV: A, K # 0; X2 > 41~1 In this case equation (28) becomes

- t+)(v- t - )

=v (l.I

or equivalently

+c =

JKJt

J

(t++t- )-I

(v - t+)(v

- t-)

with

(t++ t J 1 This gives

291

=

1x4 -

-

v - t+ v - t-

e A t e w l n l = ___

and the condition v(0) = 0 fixes eAc/lnl = At

e

5 t-

t+(v - t - ) = t-(v - t+)($ (eA t t+ - t-)v = eA t t+t- - t+tv=

eAt - 1

1

- t-

e%+

eAt- 1 l l ~ le + - t-

t+t- = -

Att

In conclusion

v(r) =

eAT- 1 eATr+

where

2

-r* := 2 (1 f

- r-

/-).

Class V: X , K # 0; X2 < 4161 The result is the same as in the case of Class IV, i.e. (59).

12. Solutions of the equation for f Class I X=r;=O.

In this case

and the condition

f ( 0 )= 1 is satisfied.

Class I1 Let us consider the case R f O

292

(59)

A2 = 4(n(. In this case we have

A#O;

&=o

then

Therefore in case (64)

x f(t) =e

t y

(;

-t)

!?+-A)

Classes IV and V These classes are characterized by the condition

x2 # 4 1 ~ 10 f

(65)

(real roots, class IV; complex conjugate roots, class V). In this case the characteristic polynomial 1 - At - fit2 has 2 different roots t f such that

Therefore the solution of equation (26)is

The solution of (67) satisfying 0 = In f(1)= In 1 293

is given by

It is convenient to write

With these notations

A = A1

+ ~ z t +;

B = -(A1

+~ z t - )

(70)

and (68) becomes equivalent to (1- t/t+)

[

f(t) = [(I -

I .

(1- t/t+)t+ nz

(1- t/t+ Remark. In the paper p] (pg. 10) Meixner assumes that 11

=0

therefore, due to (69), in order to recover his expression for f ( t ) ,one has to put A1 = 0 in (71).

13. The equations for u Now notice that, if U(t)

=7

U-’(T)

=t

then 1

t = u-’(u(t)) = v ( u ( t ) )=+ 1 = v’(u(t))u’(t) H v‘(u(t))= u’(t> Therefore if ~ ’ ( 7= ) F ( v ( 7 ) )then 1

- = v’(u(t))= F ( v ( u ( t ) ) = ) F(t) u’(t) and in our case this becomes 1 = 1- Aw(u(t)) - 6 2 ( U ( t ) ) = 1- A t - 6t2

u’(t)

294

.

or

u'(t) =

1 1 - - Kt2

x

Therefore in the case (61)

u ( t )=

J

dt = t

because u(0) = 0. In the case (64)

u ( t )=

-.

Ji"",t

X # O .

1

Therefore

u(t)= In

1 (1 - At)'/X

-

In case (63), i.e.

In case (65) the polynomial (29) has two roots given by (66). Therefore

1

- tt+ 1/14 . tt -

t-t, t - t-

- - In-=ln(-) IKI

IQ=P=A=K=O

Q$(x) = e x 2 / 2 k d x . Hermite's polynomials.

IIa=P#O

295

confluent hypergeometric polynomials I11 ff # 0, p = K = 0

Charlier's polynomials IV Q! # p, rc # 0, a l p real e x . Iff1 > IPI

v ff # p, # 0 E = K

p

14. Moments of the Meixner measures In this section we derive a simple formula which expresses the moments of a Meixner measure in terms of the associated Meixner pair. Taking &derivatives of both sides of (15) one obtains

or equivalently, writing

u(")(t):= dtu(n-l)(t) ; u(O)(t):= u(t)

(72)

In particular, putting t = 0 in (73) and using (9) one finds the first moment of p, i.e.

296

Taking derivatives of both sides of (73) one finds

or equivalently

Now, considering u ( l )as a multiplication operator in L2(Iw,p ) and at as an operator in the same space, one can introduce the notation

1

A, := -Lit

(76)

U(1)

So that, if cp is another multiplication operator in L2(Iw,p ) :

1 nuP(t) = (8tcpHt) u(1)( t )

(77)

In these notations (75) becomes

s,

z2e""(t)p(dx) = A:-

1

f

(t)

(78)

which gives the second moment of p by evaluating (78) at t = 0 mz(p) = L x 2 p ( d x ) = A:- 1 ( 0 ) .

(79)

f Now suppose by induction that,

z n e x " ( t ) p ( d z ) = A:- 1 ( t )

f then, taking derivatives of both sides, one finds

xn+lu'(t)e""(t)p(dx) = &A:-

1

f

(t)

or equivalently p(dz) =

1

1

-8th:- ( t )= A:+'U(l)(t) f

1

f

(t)

and therefore (80) holds for each n E N.In particular, taking t = 0 in (80) one finds the n-th moment of u: m,(p) = L x n p ( d x ) =A,"-1 ( 0 ) ; V n E N (81)

.

f

In other words: 297

Theorem 14.1. Suppose that a probability measure p on 0% has an orthogonal generating function of the form (8) for a pair of Cw-functions ( u ,f ) from Iw to R. Then p is polynomially determined b y the pair (u,f ) through formula (81).

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