Quantum Information V: Proceedings of the Fifth Conference, Meijo University, Japan 17 - 19 December 2001

Quantum Information V Edited by T. Hida K. Saito A i^ ^ ; .• : ' ] i 1 < I World Scientific Quantum Informat...

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Quantum Information V Edited by

T. Hida K. Saito A

i^

^

;

.•

: ' ] i

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World Scientific

Quantum Information V

This page is intentionally left blank

Proceedings of the Fifth International Conference

Quantum Information V Meijo University, Japan

17 - 19 December 2001

Edited by

T. Hida & K. Saito Meijo University Japan

\jjp World Scientific NEW JERSEY • LONDON • SINGAPORE •

BEIJING • SHANGHAI • H O N G K O N G • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

QUANTUM INFORMATION V Proceedings of the Fifth Conference Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-585-1

Printed in Singapore by World Scientific Printers (S) Pte Ltd

PREFACE

The Fifth International Conference on Quantum Information was held at Meijo University in Nagoya, Japan, December 17-19, 2001. The conference covered the following topics: (1

Classical and Quantum Information Theory

(2:

Quantum Communication Gate on Fock Space

(3

Quantum Teleportation, Process of Recognition

(4:

White Noise Theory

(5

Infinite Dimensional Stochastic Analysis Interacting Fock Space obtained from Graphs

(7;

Financial Mathematics

(8

Nonlinear Diffusion Equations

This volume contains the papers of invited lectures and contributed talks at this conference. All papers to the volume have been refereed. The editors are grateful to the referees for their efficient help. This conference was supported by the Research Project "Quantum Information Theoretical Approach to Life Science" for the Academic Frontier in Science promoted by the Ministry of Education in Japan and was also supported by Meijo University. We would like to express our sincere thanks to the Faculty of Science and Technology of Meijo University for their assistance during the conference. Last not least we express our deep appreciation to World Scientific Publishers, in particular Ms E. H. Chionh as their acting representative for a remarkably pleasant coorperation. December 30, 2002 Takeyuki Hida Kimiaki Saito Meijo University

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CONTENTS

Recognition and Teleportation M. Ohya, K.-H. Fichtner and W. Freudenberg

1

Quantum Information and Spacetime Structure I. V. Volovich

19

On Gaussian and Poisson White Noises N. Asai

35

Renormalization, Orthogonalization, and Generating Functions N. Asai, I. Kubo and H.-H. Kuo

49

Insider Trading in Continuous Time E. Barucci, R. Monte and B. Trivellato

57

Existence, Uniqueness, Consistency and Dependency on Diffusion Coefficients of Generalized Solutions of Nonlinear Diffusion Equations in Colombeau's Algebra H. Deguchi

77

On Mathematical Treatment of Quantum Communication Gate on Fock Space W. Freudenberg, M. Ohya and N. Watanabe

89

A Frontier of White Noise Analysis T. Hida

103

An Interacting Fock Space with Periodic Jacobi Parameter Obtained from Regular Graphs in Large Scale Limit A. Hora and N. Obata

121

Error Exponents of Codings for Stationary Gaussian Channels S. Ihara

145

Vlll

White Noise Analysis on Classical Wiener Space Revisited Y.-J. Lee and H.-H. Shih

159

Fractional Brownian Motions and the Levy Laplacian K. Nishi, K. Saito and A. H. Tsoi

181

Jump Finding of a Stable Process Si Si, A. H. Tsoi and Win Win Htay

193

On Entropy Production of a One-dimensional Lattice Conductor S. Tasaki

203

Quantum Information V

Quantum Information V Eds. T. Hida and K. Saito (pp. 1-17) © 2006 World Scientific Publishing Co.

R E C O G N I T I O N A N D TELEPORTATION MASANORI OHYA Department of Information Science, Science University of Tokyo, Noda City, Chiba 278-8510, Japan, E-mail: [email protected] KARL-HEINZ FICHTNER Friedrich-Schiller-Unversitat Jena, Fakultat fur Mathematik und Informatik, Institut fur Angewandte Mathematik, 07740 Jena, Germany, E-mail:fichtnerQminet. uni-jena. de WOLFGANG FREUDENBERG Brandenburgische Technische Universitat Cottbus, Fakultat 1, Institut Mathematik, PF 101344, 03013 Cottbus, Germany, E-Mail: freude@math. tu-cottbus. de

fur

We study the function of brain, in particular, we try to describe the process of recognition. We propose a brain model, in which information is processed like quantum computer. In order to understand the fundamental parts of the recognition process, the quantum teleportation scheme and its channel expression will be useful.

1

Introduction

We study the function of brain, in particular, we try to describe the process of recognition. In order to understand the fundamental parts of the recognition process, the quantum teleportation scheme 3,2 ' 6,7 is useful. We consider a channel expression of the teleportation process that serves for a simplified description of the recognition process in brain. It is the processing speed that we take as a particular character of the brain, so that the high speed of processing in the brain is here supposed to come from the coherent effects of substances in the brain like quantum computer, as was pointed out by Penrose. With those in our mind, we propose a model of brain describing its function as follows: The brain system BS =X is supposed to be described by a triple ( B(H), S{H) , A*(<5) ) on a certain Hilbert space W,where B(H)is the set of all bounded operators on H, <S(7i) is the set of all density operators and A*(G) is a channel giving a state change with a group G. Further we assume the following: {l)BS is described by a quantum state and the brain itself is divided into

2

several parts, each of which corresponds to a Hilbert space, so that H =®k^-k and ip = ©fc^fc, Xo, H =Hp®HoThus in our model, the whole brain may be considered as a parallel quantum computer 9 , but we here explain the function of brain as a quantum computer, more precisely, quatum communication process with entanglements like quantum teleportation process. We will explain the mathematical structure of our model. Let be s = {s1^2, • • • ,sn} a given (input) signal (perception) and be s = {s1, s2, • • •, s"} the output signal. After the signal s enters in brain, each element s J of s is coded into a proper quantum state pi £ S (Tip), so that the state corresponding the signal s is p = <8)jPj. This state may be regarded as a state processed by brain and it is coupled to a state stored po as a memory (pre-conciousness) in brain. The processing in brain is expressed by a quantum channel A* (or Ap A0) properly chosen by the form a state

memory

o

channel /

fn&H$n.

71=0

It is important to notice that there exist unitary isomorphisms Ux between (L2)x, X = G,P, and T(Hc) determined uniquely by the exponential functions (vectors)

^(aO^expfM-^l2]

~nT n=0

e(0,

£e£C

(3.1)

40

when X = G and f (x) = expf(i,log(l + 0> - f mdt]

<— e ( 0 ,

£ € £c

(3-2)

when X = P, respectively. It should be noted that the Poisson case has not been paid attentions so far to discuss characterization thoerems by many authors 3 - 4 - 5 ' 7 ' 9 ' 10 ' 18 ' 16 ' 17 ' 21 - 25 . It is known that {f ;£ G Ec] spans a dense subspace of [i?]«,x and {e(£);£ G Ec} does the same for Tu(Ec)- In those cases, it holds that the (L 2 )x-norm of

|M|2 = / The Sx-transform

2 \ ! | / „ | 2 = ll(/n r(tfc)-

of ip G (L2)x> given by

(Sx
[

z{xWx{x),

£G£C,

JB'

is an isomorphism from (L2)x onto the Hilbert space K of holomorphic functions F on .Ec with a reproducing kernel exp[(£,rj)\, £,TJ £ E. Let [EPUx = { 0}. Hence it can be shown easily that [^J u ,x C {L2)x by the condition (G2), and [-£]«,x is a nuclear space. The dual space [£]* x is called the space of generalized functions. Then we obtain the following continuous inclusions: [E}u,x

- > [Ep]u,x

-

(L2)x

<-+ \Ep]l -

[E]*u<x,

2

and [E]U:x C (£ )x C [E]*uX is a Gel'fand triplet. Since u G ^+,1/2 satisfies (G3), the exponential function (f>f(x) G [-E]u,x for any £ G EQ- Hence the Sx -transform can be extended to a continuous linear functional on [E]^ x as follows. Definition 3.1 (Sx-transform). For $ G [E]*uX, Sx-transform is defined by

(5Jr*)(O = ((*,0f)>,

^£c,

(3.3)

where ((•, •)) is the bilinear pairing of [£]* x and [£]«,*• • Now we come to the characterization of [£]* x associated with px, X = G, P, in a single statement. Theorem 3.2. Let a measure fix on E* be given. Suppose u G C+,1/2 satisfies conditions (G1)(G2)(G3). Then a C-valued function F on Ec is the Sx-transform of a generalized function in [£]* x if and only if it satisfies the conditions:

41

(a) For any £, rj € EQ, the function F(z£ + rj) is an entire holomorphic function of z e C . (b) There exist constants K,a,p>0

such that

\F(0\
VteEc.

On Gaussian and Poisson White Noises

In the previous two sections, we have constructed the Gel'fand triples in terms of Fock space and Schrodinger representations with relevant transformations, J and Sx- Notice that in the Section 2, we did not introduce any probability measure on E* as the standard white noise theory 19 ' 20,23 . Hence, the property of a measure on E* plays virtually no role in the definition of J-transform. In fact, the essential tools to prove Theorems 2.2 and A.3 are the Cauchy integral formula for entire holomorphic functions of several variables, Legendre transform, dual function, Schwartz kernel theorem, and properties of the nuclear space. • So, does it imply that the considerations of the growth order of holomorphic funcions and associated topologies are sufficient to examine stochastic processes by the generalized function theory on infinite dimensional space ? = > The answer is completely No even for the study of fundamental stochastic objects such as Wiener and Poisson processes. Let us emphasize the following point. It is easy to see that the "flow"
*

f(0,l [ 0 ,t],0,---) \(0,-lM],0,---)

ift>0 ift<0,

is an element of T(Hc)- Then the "tangent vector" $>t is (0,S t ,0, •••) and belongs to TU{EC)* with u(r) = e r and J$t(0 = £(*)• On the other hand, it is known that the Brownian motion B(t) is represented by [)

\-i?(im)

if*
Similarly, the compensated Poisson process is given by

[-fiih.ow

ut<°-

[

'

42

Since characteristic functions ljo^j and l[ti0] are elements of H, B(t) and P(t) — t are in (L2)G and (L2)p, respectively. Hence we obtain UG^UQ1 = B(t) and UP$tUpl = P{t) - t. So the distributional derivative of B{t) with respect to t, so-called Gaussian white noise B(t), has the form B(t) = lf{5t) for each ( 6 E . Similarly, the Poisson white noise P(t) has the expression Pit) r 1 = l[(St) for each t G R. In those cases, since St is in i?*, B(t) and P(f) — 1 belong to [£]* G and [J3]* p , respectively (A function u will be chosen in the proof of Theorem 4.2). The relationships between the vector $t, classical Gaussian and Poisson white noises have been discussed by AsaiKubo-Kuo 8 as follows. Proposition 4.1. It holds that (1) UG$tUGl = B{t), (2) Up4tUp1 = P(t)-l. It can be shown by Theorem 2.2 that the tangent vector <$t of the flow t is the tangent vector corresponding to Wiener process or (compensated) Poisson process. So it is quite natural to focus on the noncommutativity on the tangent space. In quantum stochastic calculus 11,24 , at + a* is called the quantum Gaussian white noise and at + a* + Nt + I is called the quantum Poisson white noise, which act on the same generalized functions space Tu(Ec)*- Quantum Gaussian and Poisson white noises have the different Lie algebraic nature as follows. Facts • Operators at,a*,I fact, we have

are generators of the Heisenberg-Weyl algebra Qhw In [as,a*]

=Ss(t)I,

[a„at} =0, [a*,a*} = 0 as in (2.4). • Operators at,a*,Nt,I we have

are generators of the oscilator algebra gos. In fact,

[as, a*] =Ss(t)I, [Ns,at] = -S3(t)at,

[Ns,a*t] = 8s(t)a*,

[a„at] = 0, [a*s,a*t] = 0, [N„Nt] = 0. Now, how do we relate classical white noises with quantum white noises ? By considering the isomorphism Ux, it is possible to see the classical-quantum

43

correspondences and distinguish two different kinds of white noises. This issue has been addressed in the papers by Asai-Kubo-Kuo 8 and Ito-Kubo 13 from the viewpoint of white noise theory. T h e o r e m 4.2. It holds that = (1) Uc{at +atWc ^(*)» where B(t) is considered as a multiplication operator. (2) Up(at +a% + Nt + I)Upl = P(t), where P(t) is considered as a multiplication operator. For the proof of Theorem 4.2, the essential point is to compute matrix elements for B(t) and P(t) — 1, and to notice crucial differences on product laws of exponential vectors, ^(x)^(x)=4>f+v(x),

(4.3)

# ( * ) < ( * ) =#+,+,*(*).

(4.4)

In this paper, our proof will not be based on the differential operator dttG and difference operator dt:p as in the paper 8 , but on Equation (4.3)(4.4) as follows. Proof. Let us start the proof with the Gaussian case X = G, first. (J(at + a t *)e(0)fo) = « ( « . + a j ) e ( 0 , e f o ) » r >

£,* 6 Ec

= W)+r}{t))e{i'n)-

(4-5)

On the other hand, since we have Equation (4.3) and (SGB)(£) = (St,0 = £(t) satisfies the condition (b) with u(r) = exp(r) in Theorem 3.2, (SGBf)(r,) = ((B(t),4>f4>°))G = m)+v(t))e<^K

(4.6)

Due to Equations (4.5)(4.6), we have Uc(at + OI)UQ1 = B(t), where B(t) is considered as a multiplication operator. Hence we get B{t) £ (E)Q. Therefore, we have finished to prove our first assertion. Next consider the Poisson case X = P. (J(ot + a*t + a*tat + /)e(0)fa) = <<(«* + a* +Nt + J ) e ( 0 , e f o ) » r > = m)+v{t)+v(mt)

+ l)e<^

^

€ EC

(4.7)

Note that the function rj£ above makes sense as a member of E
44

satisfies the condition (b) with u(r) — exp(r) in Theorem 3.2. So Equation (4.4) gives (SP(P - l ) 0 f )fo) = « P ( t ) - 1, # < »

p

= m+v(t)+vmt))e&").

(4.8)

1

By Equations (4.7)(4.8), we have UP(at + a;+Nt + I)Up = P(t), where P(t) is considered as a multiplication operator. Hence we get P[t) e (.E)p. Thus we have proved the second claim. • Appendix A

On Characterizations of Generalized and Test Functions

Remark A.l. Theorem 3.2 was first proved by Potthoff-Streit 25 in case of X = G and u(r) = er. It was extended to the case of X — G and u(r) = exp[(l — j3)rT^] by Kondratiev-Streit 16 ' 17 . Moreover, Cochran et al. 9 proved the case when X = G and the growth condition (b) is determined by the exponential generating function Ga{r) = ^ ^f-rn. Asai et al. 4 , 6 ' 7 minimized conditions on sequences {a(n)} of positive real numbers in such a way that Theorem 3.2 holds. Example A.2. The Gel'fand triplet [E]u>x C {L2)x C [E]*uX becomes (1) the Hida-Kubo-Takenaka space19'20'23 if X = G and u(r) = er, and the Ito-Kubo space12 if X = P and u(r) = er, (2) the Kondratiev-Streit space17 if X = G and u(r) = exp[(l + /3)rT+?] for 00,

there exists a constant K > 0 such that

|F(0| < ffu(a|£|ip)1/2' V^Ec-

45

The characterization of [£^]«,x associated with fix, X = G,P, is stated below in a single statement. The proof associated with JIG can be found in our paper 7 . Theorem A.4. Let a measure fix on E* be given. Suppose u £ C+,1/2 satisfies conditions (G1)(G2)(G3). Then a C-valued function F on Ec is the Sx-transform of a generalized function in [E]Uix if and only if it satisfies the conditions: (a) For any £,77 € Ec, the function F(z^ + 77) is an entire holomorphic func-

tion o / z 6 C , (b)' For any constants a, p > 0, there exists a constant K > 0 such that \F{i)\ < KuiaWl,,)1/2,

V£ G Ec.

Remark A.5. Theorem A.4 was proved by Kuo et al. 21 in case of X = G and u(r) = eT. It was extended to the case of X = G and u(r) — exp[(l -f-/?)/"^] by Kondratiev-Streit 17 . Moreover, Asai et al. 3 proved the case when X = G and the growth condition (b)' is determined by the exponential generating function Gi/a(r) = Y2 STnTnT7"™- -A-sai e ^ al. 4,6 ' 7 minimized conditions on sequences {a(n)} of positive real numbers in such a way that Theorem A.4 holds. References 1. N. Asai, A note on general setting of white noise analysis and positive generalized functions, RIMS Kokyuroku (Kyoto) 1139 (2000), 19-29. 2. N. Asai, I. Kubo, and H.-H. Kuo, Bell numbers, log-concavity, and logconvexity, Acta. Appl. Math., 63 (2000), 79-87. 3. N. Asai, I. Kubo, and H.-H. Kuo, Characterization of test functions in CKS-space, in: "Mathematical Physics and Stochastic Analysis: in honor of L. Streit", S. Albeverio et al. (eds.) World Scientific, (2000), pp. 6878. 4. N. Asai, I. Kubo, and H.-H. Kuo, CKS-space in terms of growth functions, in: "Quantum Information II", T. Hida and K. Saito. (eds.) World Scientific, (2000), pp. 17-27. 5. N. Asai, I. Kubo, and H.-H. Kuo, Characterization of Hida measures in white noise analysis, in: "Infinite Dimensional Harmonic Analysis", H. Heyer et al. (eds.) D. M. Grabner, (2000), pp. 70-83. 6. N. Asai, I. Kubo, and H.-H. Kuo, Roles of log-concavity, log-convexity, and growth order in white noise analysis, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 4 (2001), 59-84.

46

7. N. Asai, I. Kubo, and H.-H. Kuo, General characterization theorems and intrinsic topologies in white noise analysis, Hiroshima Math. J., 31 (2001), 299-330. 8. N. Asai, I. Kubo, and H.-H. Kuo, Gaussian and Poisson white noises with related characterization theorems, preprint (2002). 9. W. G. Cochran, H.-H. Kuo, and A. Sengupta, A new class of white noise generalized functions, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1 (1998), 43-67. 10. R. Gannoun, R. Hachaichi, H. Ouerdiane, and A. Rezgui, Un theoreme de dualite entre espaces de fonctions holomorphes a croissance exponentiele, J. Funct. Anal., 171 (2000), 1-14. 11. R. L. Hudson and K. R. Parthasarathy, Quantum Ito's formula and stochastic evolutions, Comm. Math. Phys., 93 (1984), 301-323. 12. Y. Ito, Generalized Poisson functionals, Prob. Th. Rel. Fields, 77 (1988), 1-28. 13. Y. Ito and I. Kubo, Calculus on Gaussian and Poisson white noises, Nagoya Math. J., I l l (1988), 41-84. 14. Yu. G. Kontratiev, Nuclear spaces of entire functions in problems of infinite dimensional analysis, Soviet Math. Dokl., 22 (1980), 588-592. 15. Yu. G. Kondratiev, P. Leukert, and L. Streit, Wick calculus in white noise analysis, Acta Appl. Math., 44 (1996), 269-294. 16. Yu. G. Kondratiev and L. Streit, A remark about a norm estimate for white noise distributions, Ukrainian Math. J., 44 (1992), 832-835. 17. Yu. G. Kondratiev and L. Streit, Spaces of white noise distributions: Constructions, Descriptions, Applications. I, Reports on Math. Phys., 33 (1993), 341-366. 18. I. Kubo, H.-H. Kuo, and A. Sengupta, White noise analysis on a new space of Hida distributions, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2 (1999), 315-335. 19. I. Kubo and S. Takenaka, Calculus on Gaussian white noise I, II, III, IV, Proc. Japan Acad., 56A (1980), 376-380, 56A (1980), 411-416, 57A (1981), 433-437, 58A (1982), 186-189. 20. H.-H. Kuo, "White Noise Distribution Theory", CRC Press, 1996. 21. H.-H. Kuo, J. Potthoff, and L. Streit, A characterization of white noise test functionals, Nagoya Math. J., 121 (1991), 185-194. 22. Y.-J. Lee, Analytic version of test functionals, Fourier transform, and a characterization of measures in white noise calculus, J. Funct. Anal., 100 (1991), 359-380. 23. N. Obata, "White Noise Calculus and Fock Space", Lecture Notes in Math. 1577, Springer-Verlag, 1994.

47

24. N. Obata, Generalized quantum stochastic processes on Fock space, Publ. of RIMS Kyoto Univ., 31 (1995), 667-702. 25. J. Potthoff and L. Streit, A characterization of Hida distributions, J. Funct. Anal., 101 (1991), 212-229.

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Quantum Information V Eds. T. Hida and K. Saito (pp. 49-55) © 2006 World Scientific Publishing Co.

RENORMALIZATION, ORTHOGONALIZATION, A N D GENERATING FUNCTIONS NOBUHIRO ASM, IZUMI KUBO, AND HUI-HSIUNG KUO

ABSTRACT. Let fj. be a probability measure on the real line with finite moments of all orders. Apply Gram-Schmidt orthogonalization process to the system {1, x, • • • , x",... } to get a sequence {Pn}^L0 of orthogonal polynomials with respect to (i. In this paper we explain a method of deriving a generating function ip(t, x) for fi. The power series expansion of i/)(t, x) in t produces the explicit form of polynomials P n , n > 0. 1. GAUSSIAN MEASURE AND H E R M I T E POLYNOMIALS

Let /i be the Gaussian measure on the real line with mean 0 and variance a2, dn(x) = - ^ L - e - * 2 / 2 " 2 dx. V27roIt is well-known that the Hermite polynomials [n,2]

„i

H

^^=E2kkl(nl2ky^2)k^2k, k=0

v

(1.1)

n = 0,l,2,...,

(1.2)

'

form an orthogonal basis for the space L2((i). generating function et*-«H*,2

Moreover, we have the well-known

= Y;-yHn{x;o2).

(1.3)

n=0

Now we raise the questions: (1) How to derive the Hermite polynomials from the Gaussian measure fi? (2) How to derive the generating function il>(t, x) = etx~" * I2 from the Hermite polynomials or the measure /i? Of course, one can say that the Hermite polynomials can be derived by applying the Gram-Schmidt orthogonalization process to the system {1, x, x2,... , xn,... }. But this derivation is impractical since it does not give the explicit form of the n-th Hermite polynomial Hn(x;cr2). On the other hand, suppose we know the answer to the second question, then we can just expand the generating function as a power series in t to obtain the Hermite polynomials. Therefore, the real problem is to find the generating function 4>(t,x) = etx~° ' I2 in Equation (1.3). The idea to find an answer to this naive problem comes from the multiplicative renormalization introduced by Hida in his book [5]. Consider the function
(1.4)

(The issue of how to find this function will be addressed in Section 3.) Regard x as a random variable with distribution given by n in Equation (1.1) and take the

50 expectation to get Then define the multiplicative renormalization of ip(t, x) by

««•*>- ^

k ""-'""•

(1 5)

'

Thus we have derived the generating function in Equation (1.3) by the multiplicative renormalization of the function in Equation (1.4) with respect to the Gaussian measure fi. Note that ip(t, x) is a smooth function of t and so has a series expansion

w.*) = E ^ * n -

(!-6)

n=0

On the other hand, il>{t,x) can also be expanded as 4>{t,x) =

etxe-aH*>2

- ( 5Z „!*")( 5Z 2"n! t2n) =

SlS("-2*)! 2**! J'""

(L7)

Prom Equations (1.6) and (1.7) we see clearly that Pn(x) is exactly the Hermite polynomial in Equation (1.2). Thus we have derived Hermite polynomials from the Gaussian measure fi via the functions ip(t, x) in Equation (1.4) and ip(t,x) in Equation (1.5). Moreover, it can be easily checked that for any t and s, E^(t,-)ij(s,-) = e^ts. The fact that Efj.ip(t, -)ip(s, •) is a function of the product ts implies that the Hermite polynomials are orthogonal with respect to the Gaussian measure (j,. 2. POISSON MEASURE AND CHARLIER POLYNOMIALS

Next we consider the Poisson measure with parameter A > 0,

c(W) = e " 4

* = 0,1,2

(2.1)

How to proceed to find a complete orthogonal sequence of polynomials for fi and the corresponding generating functions? Consider the function V{t,x) = {l+tf.

(2.2)

(The issue of how to find this function will be addressed in Section 3.) Regard x as a random variable with distribution given by fj, in Equation (2.1) and take the expectation to get E^{t,-)=ext. Hence the multiplicative renormalization of
w>x^i$$r)=*-"<1+tr-

(2 3)

-

51 This function can be expanded as a power series in t as follows:

*..>-(£^)(i*K x

n=0

n=0

'

v

V

fc=0

v

n=0

>

'

'

where px$ = 1 and px>n = x(x — 1) • • • (x — n + 1) for n > 1. In view of Equation (2.4) we define the n-th CharUer polynomial Cn(x;X) by C n (x;A) = £ ( £ ) ( - A ) " - W -

(2-5)

Then we have the equality 00

1>{t, x) = e -

At

fn ( l + t) = Y, ^jCW(i; A). x

(2.6)

n=0

Moreover, we can easily check that for any t and s, = ext°.

E^(t,-W(s,-)

Since E,j.ip(t, -)il>(s, •) is a function of the product ts, the Charlier polynomials are orthogonal with respect to the Poisson measure fj,. Thus the function ip(t,x) in Equation (2.3) is a generating function for the Charlier polynomials. 3. G E N E R A L CASE

Now, consider a general probability measure [/, on the real line R. Assume that J"K |x|" dn(x) < oo for all n = 0,1,2, Then we can apply the Gram-Shcmidt orthogonalization process to the system {1 « 30* 00 * * • • « 00 « * • • } to get a sequence {P„; n = 0 , 1 , 2 , . . . } of orthogonal polynomials such that PQ = 1 and Pn is a polynomial of degree n with leading coefficient 1. Q u e s t i o n : How to find the explicit form of the polynomial

Pn(x)?

Being motivated by the examples in the previous sections, we consider a function of the form oo V{t,x)

= Y,9n{x)tn, n=0

where g„(x) is a polynomial of degree n satisfying the condition limsupHsnll^" < o o . n—>oo

v f v

Let tp(t, x) be the multiplicative renormalization of (p(t, x) defined by

Then we have the following fact. For details, see the papers [1] and [2].

(3.1)

52 Theorem: The function E^iplt, -)ip(s, •) is a function of the product ts if and only ifip(t, x) is a generating function for fi, i.e., it has the series expansion

f^Qn(x)tn,

i>(t,x) =

(3.3)

n=0

where Qn{x) is a polynomial of degree n and the polynomials Qn 's are orthogonal with respect to the measure p,. Let an be the leading coefficient of Qn. Then the polynomials we are looking for in the above question are given by Pn(x) = Qn(x)/an. Moreover, we have the equality oo i>(t,x)

YtanPn(x)tn.

= n=0

Thus here is the A n s w e r to the above Question: Find a function (s, •) is a function ofts. Then the power series expansion ofip(t,x) in t will produce the polynomials P„{x). But then the key point is to find such a function ip{t,x). At the present time we can handle functions of the following type: 1. x with p(t) being determined by the above theorem. Here are some examples:

Gaussian N(0,cr2), a > 0 Poisson Poi(X), A > 0 Gamma T(a), a > — 1 2. (f(t,x) = (l — p{t)x)

ip(t,x)

i>(t,x)

etx

e

(l + t)x

e~xt(l+t)x

tx

Pn(x)

ti-itr2t2

t)-01-^^

(l +

Hermite Charlier Laguerre

with p(t) and c being determined by the above theorem.

Here are some examples: M Uniform on [—1,1] r(/?+1)

x2)3-*

. (1

|x|
(/3>-i,/3^0)

T Vl-X2 '

\x\ < 1

ip(t,x)

Vl-2tx+t2

^l-2tx+fi

2

_(l+t if_

1

(l-2tx+P)t>

(l-2tx+t 2 )P

4-4tx+t2

4-t2 4-4tx+t 2

Pn(x)

Legendre Gegenbauer Chebyshev (first kind)

\VT

\x\<\

l-2tx+t*

l-2tx+t*

Chebyshev (second kind)

Note that the Legendre and Chebyshev of the second kind are the special cases of Gegenbauer with f3 = 1/2 and /3 = 1, respectively. However, although the measure for the Chebyshev of the first kind is a special case of the Gengenbauer with (3 = 0, we cannot obtain the corresponding
53 of Gegenbauer by letting (3 = 0. Moreover, we want to point out that in some books (e.g., page 25 in [3]) the generating function of the Chebyshev polynomials of the first kind is stated in the form:

n ? ^ = £*.<*)*".

^

71=0

where the Chebyshev polynomial Tn(x) of the first kind is defined by Tn(x) = cos(narccosx),

n > 0.

However, our Chebyshev polynomials of the first kind in the above chart are defined (through our method of deriving Pn's from the generating function) by 4-t2 4 - 4te + 1 2

= £f n (*)*".

(3.5)

n=0

By expanding the left-hand side of Equation (3.5) as a power series in t, we can easily check that the polynomials Tn(x) are given by:

fo(x) = 1, i

T„(x) = ^

[[n/2] n/2

J

/

\

2 k

g(-l)*( -•x: ) , 2 \"z.f,'. fc )*»--(l — i

n > l .

The polynomials Tn and Tn are related by Tn(x) = 2n-1fn(x),

T0(x) = fo{x) = 1,

n>\.

With this relationship we see that Equation (3.5) implies Equation (3.4). We mention that another formulation of the generating function for Tn(x) is given by (see e.g., page 89 in [6]) 1-t2 l-2tx + t2

= T0(x) + 2j2Tn(x)tn. n=l

For the derivation of the generating functions in the above chart and further information, see our papers [1] and [2]. Below we give a new example of generating function and the corresponding orthogonal polynomials. Consider the negative binomial distribution fi with parameters 0 < p < 1 and r = 1,2,... given by

K{k}) = ( 7 ) (-1) V ( l - P)k, * = 0,1,2,... . Try the function of the first type, i.e.,
q=

l-p.

Hence the multiplicative renormalization of

(K) = n} .

Let M be a u-algebra generated by Mx,n- F is the a-finite measure on (G, Q) defined by

F{Y) = i y (p0) + Y, A / !y f I > A »n (rfzi • ••<**»). n = l n'

JG

\j=l

"

/

where l y is the characteristic function of a set Y, ip0 is an empty configulation in M and 6X. is a Dirac measure in Xj. M = L2 (M,M,F) is called a (symmetric) Fock space. We define an exponetal vector exp. : M —> C generated by a given function g : G —+ C such that [1

(

Let / and g be functions from G to C and
ex

P / (&) • exPg (
exp/.p (
97 One can observe that exp g € M if and only if g £ L2 (G, v). Vc : dom(Vc) —> A42 is the compound Hida-Malliavan derivative 2 , 3 given on a dense domain dom (Dc) containing the exponential vectors by X>c* (

,iPl,ip2eM).

In particular, for an exponential vector exp„ with g G L2 (G, v) one has D c exp 5 = expg ® exp p . The compound Skorohod integral Sc : dom (Sc) —> .M is given on the dense domain dom (5 C ) C .A4 ® A1 containing tensor products of exponential vectors by

S c $ ( v ) = 5 ^ $ (£>, V - (p) ($ e dom (5 C ), y> € M). In particular, for tensor products of exponential vectors exp g ® exp h we have Sc (exp 9 exp^) = e x p s + h

(g, h G L2 (G, v)) .

Let T be a linear operator on l? (G, v) with ||T|| < 1. Then the operator r (T) called second quantization of T is the bounded operator on Ai satisfying T (T) exp s = exp T g . Clearly,

r(T 1 )r(T 2 ) = r(T 1 r 2 ), r (r*) = r (T)*. For a function * : M —» C we denote by O^ the multiplication operator by * , i.e., 0$9 (
beam splittings

€M,
on Fock space

Let a, j3 be measurable mappings from G to C satisfying a \a(x)\2

+ \P(x)\2=l,

(xeG).

We intoduce an unitary operator Vayp, V* ^ : M ® M -^ M ® M defined by Va,0 = (Oexpa ® 0«x P/J ) ^ ,

98 Then one can obtain

{Va,p$) (2) = ^2

Yl

exp

« (^i) exP/3 (Vi ~ ^1)

x exp_y5 (£ 2 ) exp a {
* (^1 +
for $ e M ® X and tp^tpz € M. Let .4 s i (W) be the set of all bounded operators on M. and S (A) be the set of all normal states on A. £a,p '• A ® A ->.A ® .A defined by £»w8 (C) = VlpCVa,p,

VC€A®A

is the lifting in the sense of Accardi and Ohya £a,p given by

l

and the dual map £* g of

is the CP channel from &{A® A) to &{A® A). Using the exponetial vectors, one can denote a coherent state 0* by 0f (A) = ( e x P / , Aexpf)

e"11'1'2,

V/ e L 2 (G, 1/), VA € A

In particular, for the input coherent states rj0 wo = 0* ® 09, two output states wi (•) = 770 (g) w0 (£a,/3 ((•) ® -0) and T}X (•) = r?0 ® u0 (£Q)/3 (I ® (•))) are obtained by

£* ^ is called a generalized beam splitting on Fock space because it also hold the same properties satisfied by the generated beam splitting II* in Section 1. Now we introduce a self-adjoint unitary operator U, which denotes a new device instead of the Kerr medium, defined by £($)(^2)EE (-1)1^1

$ ( ^

2

)

for 6 M ® M and ip1,tp2e G, where \
^2 (-A) = wi ® K (U(A®I)U") = •±2 \m\

[ JM

F{dv)\1>{
99 for any A G A, i> £ M (ip ^ 0) and / G L2 {G, v). If K is given by the vacuum state 6 , then the output state w2 is equals to uj\ and if K is given by one particle state, that is, K(-) = jhp {ip, mip) with ip f^f (where Mj is the set of one-particle states), then LO2 is obtained by 9~. Let M0 (resp. M e ) be the set of

Wl G A

where Ai and A2 are given by /AI = PF/M0^(^)IV'MI2,

lA2 = 1 ^ / M e F ( ^ ) | V ' M I 2 . Two output states 0/3 (•) = a;2 <8> % (£a2,02 ((") ®-0) a n d % (') = w2 ^2 (^a2,/32 (-^ ® ('))) °^ the total logical gate including two beam splittings £ak,0 w ' t n (lafcl + l/'fel = l ) (^ = 1-2) and the new device instead of Kerr medium are obtained by UJ3 = \l0a^-(arf+Pl9))+02(-^lf+aig) TJ = \lO-^2(-(cnf+01g))+a2(-01f+a1g)

^2Qa2(a1f+l31g)+P2(-01f+alg)

+ +

X2Q-02(cif+01g)+a2(-01f+a1g)

where w2 = A i 0 ~ ( a i / + / 3 l 9 ) + A 2 0 Q l / + / ? l 9 and % = »h = 0 " ^ / + a i 9 . 3.2. Complete

truth table for the new logical gate

In this section, we show the complete truth table giving by the above logical gate on Fock space. We put u>i = 6* and rj1 = 09. If we assume the case (1) of Ai = 0 and A2 = 1, then one has W3 = „

_

Q(aia2-0102)fH»201+&i02)gj 0(-aij32-"2/3i)/+(-/M2+"i"2)s

and if we assume the case (2) of Ai = 1 and A2 = 0, then one has Uj3

—

(){-aia2-0102)f+{a201+a102)g

„

_

Q(ai02-a20i)f+(~0102+ociO'2)g_

For example, we have the complete truth tables for the following two cases (I) and (II): (I) When a\ = a 2 = P\ = P2 = -4= are satisfied, two output

100 states of the new logical gate become UJ3 = 83 and r}3 = Q~* under the case (1) and u>3 = 0~' and r)3 = 99 under the case (2). (II) When a^ — ^-J- and Pk = ^ w i t n lk,&k& [0, 2TT] hold ajo^ = /?i^2> o n e h a s 7i +72 = ^2 -<5i and two output states of the new logical gate become LJ3 = 69 and rj3 = 9~' under the case (1) and W3 = 6~' and rj3 = 69 under the case (2). The new logical gate treats three initial states u0, r]0 and K corresponding to two input gates I\, I2 and the control gate C, respectively. The true T and false F of the input state UJ (resp. rj) are described by two different states UJT and wF (resp. r)T and r)F), that is; True O coherent state UJT — 6* (resp. r)T = 89), False <=> vacuum state.w^ = 0° (resp. rjF =9°). Moreover, the truth state KT and the false state KF are denoted by the control states of the case (1) and (2), respectively. When the initial control state K is KF under the above case (I) or (II), the final states of the new logical gate corresponding to two input gates 0\, O2 are obtained as the following truth table:

h

h

T F T F

T T F F

C F F F F

0, T F T F

O2

T T F F

When the initial control state K is KT under the above case (I) or (II), the final states of the new logical gate corresponding to two input gates 0\, O2 are obtained as the following truth table:

h

h

T F T F

T T F F

C T T T T

Oi

O2

T T F F

T F T F

It means that the new logical gate performs the complete truth table. Further results will be appear in our joint paper 12 .

101

References 1. L.Accardi and M.Ohya, Compound channels, transition expectations, and liftings, Appl. Math. Optim., 39, 33-59, 1999. 2. K.H. Fichtner and W. Freudenberg, Remarks on stochastic calculus on the Fock space, Quantum Probability and Related Topics VII, World Scientific Publishing Co., 305-323, 1991. 3. K.H. Fichtner, W. Freudenberg and V. Liebscher, Beam splittings and time evolutions of Boson systems, Fakultat fur Mathematikfi und Informatik, Math/ Inf/96/ 39, Jena, 105, 1996. 4. E.Fredkin and T. Toffoli, Conservative logic, International Journal of Theoretical Physics , 21 , 219-253, 1982. 5. G.J. Milburn, Quantum optical Fredkin gate, Physical Review Letters , 62, 2124-2127, 1989. 6. M.Ohya, Quantum ergodic channels in operator algebras, J. Math. Anal. Appl., 84, 318 - 327, 1981. 7. M.Ohya, On compound state and mutual information in quantum information theory, IEEE Trans. Information Theory, 29, 770 - 777, 1983. 8. M. Ohya, Some aspects of quantum information theory and their applications to irreversible processes, Rep. Math. Phys., 27, 19 - 47, 1989. 9. M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer, 1993. 10. M. Ohya and N. Watanabe, Construction and analysis of a mathematical model in quantum communication processes, Electronics and Communications in Japan, Part 1, 68, No.2, 29-34, 1985. 11. M. Ohya and N. Watanabe, On mathematical treatment of optical Fredkin Toffoli - Milburn gate, Physica D, 120, 206-213, 1998. 12. W. Freudenberg, M. Ohya and N. Watanabe, Generalized Fock space approach to Fredkin - Toffoli - Milburn gate, SUT preprint.

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Quantum Information V Eds. T. Hida and K. Saito (pp. 103-119) © 2006 World Scientific Publishing Co.

A F R O N T I E R OF W H I T E NOISE ANALYSIS

TAKEYUKI HIDA Meijo University Nagoya 468-8502, Japan E-mail: [email protected]

White Noise Analysis, both classical and quantum, has developed extensively in these years, however it has been focussed mainly on the case of Gaussian white noise. Now, it seems to be a good time for us to pay more attention to non Gaussian white noise. There is, of course, much similarity between Gaussian and non Gaussian cases, but we are interested in the analysis that depends on dissimilarity between them. A significant direction will be proposed in this paper, aiming at the path-wise analysis of functionals of additive processes which are elemental processes.

AMS Subject Classification (2000) 60H40 1. Introduction Having had a very quick review of the analysis of (Gaussian) white noise functionals, A Frontier of White Noise Analysis will be proposed. When the white noise theory is discussed, functionals in question are defined on the measure space (E*, ii) of white noise, where E* is the space of generalized functions on R and /i is the standard Gaussian measure, called the white noise measure. The Hilbert space (L2) = L2(E*,(i) used to be taken as the starting point and at the same time it is considered as a milestone. To see the idea of our approach, we may say some words. If we are allowed to take a reluxed view of mathematical rigor, we will express functions and operations in the visualized forms in terms of B(t), that is white noise, to have representations of random complex phenomena. Then, one can think of how to analyze given functionals with variables 5(t)'s. Our aim is to investigate mathematically random evolutional complex systems by using the analysis of functionals of white noise, in particular we

104

shall discuss those systems which are actually observed in various applications. For the study the following steps are in order: Reduction

—»

synthesis

—>

analysis

The idea is that first the innovation of the random system is constructed, then functionals of the given innolvation are to express the original random system, and finally those functionals are to be analysed. The innovation can often be constructed by the variations of the given random evolutional phenomena, and sometimes, like in the communication theory, it is given in advance. Our attention will be focussed on the analysis of functionals of the innovations. Of course, the cases where the innovation can actually be constructed are more attractive. There are other cases where the innovation is given in advance by some ways or others, and those are also important and have been well investigated. The essential part of the analysis comes from the white noise theory, which is the central way of the analysis of functionals of the innovation. There naturally arises an infinite dimensional analysis. The last step is the application of the analysis. Many applications to quantum dynamics have been known; now application to bioscience would be the most fruitful area, although only part of it will be illustrated in what follows.

2. Background 2.1.

Reduction

To fix the idea let us observe the case where the random complex system is taken to be a stochastic process X(t). Levy's stochastic infinitesima equation for a stochastic process X(t) is expressed in the form 6X(t) = $ ( * ( * ) , 3
105

contains the same information as that newly gained by the X(t) during the infinitesimal time interval [t, t + dt). If such an equation is obtained, then the pair ($, Y{t)) can completely characterize the probabilistic structure of the given process X(t). Under mild assumptions the innovation may be considered as the time dereivative of an additive process Z(t). Further it may be assumed that the additive process has no fixed discontinuity and has stationary independent increments. Tacitly it is assumed there is no non-random part. Then, the Levy decomposition of such an additive process shows that Z(t) = X0(t) +

X^t),

where Xo(t) is Gaussian, in fact, a Brownian motion up to constant, and where X\ (t) is a compound Poisson process involving mutually independent Poisson processes with various jumps. Thus, a Brownian motion and Poisson processes with various jumps are all elemental additive processes. As a generalization of the stochastic infinitesimal equation for X(t), one can introduce a stochastic variational equation for random field X(C) parameterized by an ovaloid C:

8X{C) = $ ( X ( C ' ) , C < C,Y(s),s

e

C,C,6C),

where C < C means that C" is in the inside of C. The system {Y(s), s G C} is the innovation which is understood in the similar sense to the case of X(t). The two equations above have only a formal significance, however we can give rigorous meaning to the equations with some additional assumptions and the interpretations to the notations introduced there. The results obtained at present are, of course, far from the general theory, however one is given a guideline of the approach to those random complex evolutional systems in line with the innovation theory and hence with the white noise theory. As in the case of X(t) we can consider elemental random fields, or equivalently elemental noises with multi-dimensional parameter.

106

3. Gaussian s y s t e m s 3.1. Gaussian

processes

First we discuss a Gaussian process X(t),t £T,T being an interval of R1, say T = [0, oo). Assume that it is separable and has no remote past. Then, the innovation which is to be a white noise B(t) can be constructed explicitly. The original idea of such an approach to Gaussian processes can be found in the P. Levy's 1956 paper (Proceedings of the third Berkeley Symposium). Under the assumption that the process has unit multiplicity and under other mild assumptions, the Gaussian process X(t) has innovation B(t) which is a white noise such that

X(t) = [ Jo

F(t,u)B(u)du,

where F(t, u) is a sure (non-random) kernel function. This is the so-called the canonical representation of X(t) and F(t,u) is the canonical kernel. This might seem to be rather elementary, howeveer such an understanding is not correct. Profound structure behind this formula would lead us to a deep insight as we shall see later. On the other hand, take a Brownian motion and a kernel function G{t, u) of Volterra type. And we are given a Gaussian process Z(t) expressed as a stochastic integral Z(t) = I Jo

G(t,u)B{u)du.

Assume that G(t,u) is smooth on the domain 0 < u < t < oo. Then, we have Theorem 3.1 The variation 5Z(t) of the Gaussian process Z(t) is defined and is given by 6Z(t) = G{t,t)B(t)dt

+ dt J Jo

Gt(t,u)B(u)du,

where Gt(t,u) — ^G(t,u). The B{t) is the innovation of Z(t) if and only if G(t, u) is the canonical kernel. Proof. The formula for the variation of Z(t) is obtained easily. If G is not a canonical kernel, then the sigma field B t ( X ) is strictly smaller than B t (jB),

107

in particular the B(t) is not expressed as a function of the Z(s), s < t + 0. Note that if, in particular, G(t,u) is of the form f(t)g(u), then Z(t) is a Markov process and there is always given a canonical representation. Hence B(t) is the innovation. Remark. In the variational equation for X(t), the two terms in the right hand side are of different order as dt tend to zero, so that two terms seem to be discrimiated. But in reality, the problem like that is not so simple. As a result of having obtained the innovation, we can define the partial derivative denoted by dt: dB(t)' Since it is defined by the knowledge of the process Z(s),s canonical kernel F(t,u) should be obtained by F(t,u) = duZ(t),

< t + 0, the

u
We now come to the analysis of nonlinear functionals of white noise B(t),t £ R1. The collection of those functionals with finite variance forms a Hilbert space (L2). The direct sum decomposition of (L 2 ) into the subspaces Hn,n > 0, is obtained:

(L2) = 0W„. The decomposition stands for the Fock space. The Hn is called the space of homogeneous chaos of degree n. The time propagation is particularly important. It is expressed as a one-parameter unitary group Ut,t € R, acting on (L 2 ) determined by UtB{s) = B(s +1). Appealing to the Hellinger-Hahn theorem, it is shown that Ut has 1) simple multiplicity on Hi, and 2) countable multiplicity on Hn,

n>2.

In addition we can associate a symmetric L 2 (i? n )-function with a functional

F£L2(Rn),

108

where A means symmetric. Such an isomorphism can be obtained by the S-transform: (SV)(0 = / exp[< x,£ >}if(x)d^(x), J(sy

JE-P

I

(p(x)fx(dx).

JC

The following theorem characterize the space Ep Theorem 2.3 (Lee-Shih 2 2 ). (i) For p G N, Ep consists of all the functions x £ C with the properties: (i) x,x,... ,a;(p) are absolutely continuous with x^p+1^ G L2IO, 1] and (ii) 2fc ar( )(0) = rr( 2fc+1 )(l) = 0 for k = 0 , 1 , . . . ,[p/2]. Moreover, |x| 2 = /„* \x(P+1\t)\2

dt and

(x, y) =

— / x(t) dy(t) Jo

for 1 6 I 2 and y G Ep.

(ii) The nuclear space E consists of all real-valued infinitely differentiable functions x defined on [0,1] such that x(2fc)(0) = x^ 2fe+1 ^(l) = 0 for each keNU{0}.

3 The test and generalized Wiener functionals For a fixed Banach space B with the | • |B-norm, let Bc be the complexification of B with the | • Is^-norm of Bc defined by \x + iy\sc = sup{||e i e (x-My)||B c : 9 G [0,27r]} for x,y G B, where ((x+iyUg = |Z|B + |2/IB- Now, we consider the AWS (C, C) and (C, E-p) (p > 1). Let B be either C or £ _ p . Denote by £(B) the class of those functions

(z)| < c exp{c'|z|B c } for some constants e, c' > 0. Then £(B) C L2(C,fj,) by Fernique's theorem (see, for example, Kuo 12 and the papers cited there). For m G N and

IM| £m( B)=

snp{\^(z)\e-m^:z€Bc}.

Let 5 m (B) = {

164

£{B) with the inductive limit topology induced by the family {£m(B)}. Then £{B) becomes a locally convex topological algebra. For notational simplicity, we denote £m(E-p), £m(C), £(E-P), and £{C) respectively by £m,v, £m,o, £P, and £Q. Then we have the following chain of continuous inclusions: £00 = f~lp>i £pc£pC£q
L2(C, fi)

asp>q>l,

where 1, ipn converges to

J f(x)

:hi---hn:(x)

n(dx),

(3.1)

where Dn denotes the n-th Frechet derivative in the directions of C and :hi---hn:(x)= Jc Yl%i (x + iV' hi) Kdy) (Lee 14 ). When hi = h2 = • • • = hn = x £ C, and T is n—linear Hilbert-Schmidt operator defined on C, we define Txn = L2(C,/i) - lim / T(Pk(x) + i Pk(y))n fc—00 Jc

fi(dy),

Pk(z) = Y?j=i (2> fj) fj f° r z £ &*• Then the well-known Wiener-Ito decomposition of a function / € L2(C,n) can be reformulated as follows. 00

/(*)=

E l ••Dn(^*f)(0)xn: „ n!

a.e. (/x).

n=0

Moreover,

/ \f(x)\2»(dx) = E ^ H ^ ^ X ^ W v n=0 P

Let T(,4 ) be the second quantization of Ap on and let 00

L2(C,^L)

( A = v^

7-U

)

m

for / e L 2 (C,/x). For m € N and p > 1, define the || • || m)P -norm on L2(C,n) by \\m,p

•

l|r(^)(Qm/)|U2(c,M).

165

Denote by (E)m,P the class of functions / in L2{C,p) so that ||/|| m ,p < +°°Then {{E)mjP : m £ N} is an increasing sequence of Hilbert spaces of which the inner products induced by the || • || miP -norms. Let (E)p be the inductive limit of {(£')m,p} and (E) = D p >i (E)p, equipped with the projective limit topology. Then the following chain of continuous inclusions hold:

(E)c(E)pc(E)qcL2(C,v) for p > q > 1. Note that \i * f may be extended to a function defined on £L PiC . Using this fact, we shall see that every member of (E) enjoys an analytic version. Theorem 3.1 (Lee 1 9 ). Every member f of (E) admits a unique analytic function f £ £<x> defined on E* such that f = / fi-almost everywhere on E*. In fact, /(*) =

E

f Dn(fx*f)(z

-, n

+ iy)nfi(dy),

z e E*c,

(3.2)

n=0 - JC

where the series converges absolutely and uniformly on bounded subsets of E*. The second quantization T(AP) of Ap enjoys an integral representation as follows (see Lee 19 ): for / s L2(C), we have T(AP)f(x) =

Urn / (|i * f)(A»Pk(x) + iAvpk{y)) >x(dy) fc—oo Jc

=

lim

f f f(APPk(x)+iAPPk(y)

fe—oo Jc

+

z)f,(dyMdz),fi),

Jc

where the limit is taken in L2(C,/J). Employing such a representation, we have the following growth estimates. Theorem 3.2 (Lee 1 9 , Lee-Shih 2 2 ). (i) Let f be in (E). For anyp > 1, let mp e N so that f e {E)mpiP.

Then

||/|k m p , p < Cmp \\f\\mp,p, where cm = Jc emlyl°° fi(dy) for any m £ N. (ii) Let

l, let mp € N so that ip £ £mp(E-p). Let r be a real number such that r > In 2/(2 In •K — 2 In 2), and choose rhp to be the least integer greater than y/mpe. Then IMImp.p-r < Pmp,p ||

f\\£mp,P,

where f3mp,p is a constant depending only on p and mp.

166 (Hi) SQO C (E) and the mapping f —> / is a homeomorphism from (E) onto c-oo*

Denote the dual spaces of So, £P (p > 1), and £oo respectively by ££, £*, and £^ which are topolozied by the weak*-topologies. Then we have the following chain of continuous inclusions: for p > q > 1,

5oc c £p c £q c s0 c

L2(C, M)

c £0* c £*q c e; c C -

Members of ££, will be referred as the generalized Wiener functionals (in short, GWF). Let ((•,•)) stand for the dual pairing of £Q-£Q-, £p-£P, and ££,foo, respectively, for notational convenience. The exponential vector functional £(77) associated with rj e Ec which is given by

£(77)= exp{(-,77)-i jf

fi(tfdt\.

Then e(rj) S £oo- We note that for any / e L2(C,fi), (/x* /)(??) = ((/, £(??))). Definition 3.3. TTie S-transform SF of F £ £^ is defined as a complexvalued functional on Ec by SF(rj)=

((Fein))),

n € Ec.

Denote the dual space of (E) by (E)* which is endowed with the weak*topology and let ((•, -))ym denote the dual pairing of (E)* and (E). Since (E) is dense in (E)p for any p > 1, (E) becomes the reduced topological projective limit of {(E)p). Then, by [Theorem 6, pp 290] 10 , (E)* = U p >i [E)*p which is endowed with the inductive limit topology. By Theorem 3.2, the embedding j : £(*, —» (E) is continuous. So, for any G e (E)*, G o j e ££a. Thus (E)* can be identified as a subspace of ££,. Conversely, for F G £^, define F on (E) by ((F, f))ym

:= ((F, /)),

/

G

(E),

where / is the analytic version of / on E*. It follows from Theorem 3.2 that

Fe(E)*. A functional G on Ec is analytic if it satisfies the following two conditions: (A-l) for all n,(j> £ Ec, the one complex variable mapping C 9 A \—> G(rj+X ) is entire on C; (A-2) there exists p > 1 such that for any c > 0, sup{|G(?7)| : 77 € Ec and |T7|P
167 Theorem 3.4 (The characterization theorem 2 2 ). Let F e £^ be fixed. Then the S-transform SF of F is an analytic function on Ec and there exists p>l so that for any m £ N , the number oo m 2 ™

"m,-p(f):=

E n=0

\,i-j

j-^\\DnSF(0)fHSHEp)

is finite. Conversely, suppose that G is an analytic function defined on Ec and satisfies the condition (A-2) for some p > 1. Let q be sufficiently large so that e 2 • Y^Li ^j < 1- Then there exists a unique F £ S^ such that nm-q{F) < +oo and SF = G, where Xj = (j - (1/2))TT for j £ N. For

<

||M*V>||£

\

—

11 r

I

"• / II I II c - m p . p

r l|C-mp,p

< c m \\tpWs —

""p

llr

,

M^Triptp'

where k is the least integer greater than e m p . Remark 3.6. Combining Theorem 3.3 with Proposition 3.4 we see that S(£oo) = £00c

A(EC) =

S{S^),

where A(EC) is the space of all analytic functions on Ec. Applying Theorem 3.3, one obtains Proposition 3.7. (i) SF = 0iffF = 0 for F £ £^. a total subset of £"oo.

Equivalently, the class {s(n) : n £ E} is

(it)
4 Examples of generalized Wiener functionals Let (C, B) be an AWS with B = C or B - E-p, p = 0 , 1 , 2 , . . . and let | • | B denote the corresponding norm. Unlike Hida's original approach, examples of GWF's given in this section will be defined by their linear functional forms instead of their ^-transforms, for more details we refer the reader to Lee 15 .

168

E x a m p l e 4 . 1 . Denote by Llxp(B,fi) defining on (B, B(B)) such that

the space of all measurable functions /

f(x) em ^B fi(dx) < oo

for all m G N.

/. Llxp(B,[i) is regarded as a subspace of £(B)* by identifying each / G Lg Xp (5,/x) with the functional Gf denned by

((Gf,
f(x)^x)^dx),

JB

for

1, La(B,fi) c ££,E x a m p l e 4.2. Let v be a Borel measure in (E*,B(E*)) erty that there exists p > 1 so that for all m € N, [

e™\*\-p v{dx)

satisfying the prop-

(4.1)

Then the measurable support of v is contained in E-p. define

For G £oo,

JE-v

Then v G f^,. For example, fi and Ot(x,dy) = /j,(dy — e *x) (the transition measure of Omstein-Uhlenbeck process) are GWF's in £Q . Moreover, we have Theorem 4.3 (Lee 1 8 ). Let F be a positive generalized function in £^a, that is, ((F, (p)) > 0 for (p G £oo with tp > 0. Then there exists a unique finite measure uF on (E*, B(E*)) so that {{F,
f

JE*

Moreover, the measurable support of vF is contained in E-p for some p > 1 and for all m G N, the condition (4.1) is satisfied, Example 4.4. (The generalized coordinate functionals) Let h€ B* and denote h(x) = (x, h). Then ~h G L2(fi) C ££, and

((h, V>» = / (x, h) V(x) li{dx) = (h, Dfi * V>(0)), JB

169

where Dfi * i>(0) denotes the Prechet derivative of fi. * ip at 0. For y £ B, choose {yn} C E so that yn —> y in 5 . Then, for
(x,yn)ip(x)fi{dx)

n—»oo /.->

=

lim (yn, Dfi* ip(0)) = (y, Dfi * y(0)). n—>oo

This leads the following definition:

((j7> v » = &/> -°A* * v(o)) for v? e foo. Thenyef^. In particular, the white noise B(i) associated with the Brownian motion may be defined by be defined by ((B(t),
vGSoo,

where /i t = ftbt = l [t>1] . Then 5 ( i ) € £J\ Example 4.5. (The multiplicative renormalization) Let h GC and a S C, then we have 2

e-±<*

\h\l, f c «*(*.fc> y ,( a; ) /i (d a ;) =/i¥ ,( a /i). is

Define : e Q/l := e-i Q 2 l , , £'e a ' 1 . Then we have ((:eal:,
(4.2)

(4.2) holds not only for h £ C but also for h £ B. Defining : eal : by (4.2), then : ea~h :€ <%,. In particular, let y = ht = l[t,i]. Then we have ((: exp{a5(t)} :, ip)) = (fi *
€ C and (f €

/ : hi • • • hn : (x)
•

(4.3)

Note that the right hand side of (4.3) makes sense also for hj £ B. This leads to define additive renormalization of y~i • • • yn for yj £ B given as follows: ((: 2/1 • • • Vn ••, V» = Dn{n * ip)(0)yi •••ynThen :yi---yn:£ S^. In particular, for any positive times {£1,£2, • • • , t n } C [0,1], not necessarily distinct, define the renormalization of B(ti) • • • B(tn) by ((: B(ti) • • • B(tn) :,
•••htn

170 Example 4.7. Let T be a trace class operator defined on C, then {Tx, x)o can be defined a.e.(/j) on C and / c (Tx, x)0 fi(dx) = Tr(Ts), where Ts denote the symmetrization of T. For

2(/x * [D2(w *
:€ ££,. In particular, when T = I, we have # : A B(tfdt:,

cp\

= tracer [£>2(M * ip)(0)}.

Example 4.8. (Generalized multiple Wiener integrals) For / € L 2 ([0,1]"), then we have

f In(f)(x)V(x)ti(dx)

Je

= (([

J[o,i}n

f(tx,.. .,tn)dB(t{).

f(h,..., J\z [0,1]"

..dB(tn),
tn)DnS
• • • htndh

---din (4.4)

for (p G £oo. Observe that the the mapping ^lv given by *v(*i,...,«n)=

Dn((i*ip)(0)btl---btn

lies in £ ® n . If / is a generalized function in (E*)®n, then we define

«/„(/),
{{6M

may be de-

* » = vmo eA~m) IAX~ Wofh{x) -u}h)Kdx)-

for h e C and tp € £oo. It is clearly a positive generalized function in ££,.

171

5 Calculus of generalized W i e n e r functionals • Multiplication

by test

functionals

Let F € £^ and ip e £OQ. Then for any

! define UF, F G ££,. • Translation

For z G E, F £ £*, define the translation T Z .F by
((TZF,

z e

r_^»e^lo

for

Differentiation For any z £ E and for any
((F,z
It is easy to see that DZF is the unique generalized function G in ££, satisfying SG(y) = (x, D(SF)(y)), for y € E. For any x € E*, the mapping £>x is a continuous linear operator from £<„ into itself. Let Dx denote the adjoint operator of Dx, i.e. {{D*xF,
?£ £00.

holds for r],€,&E.

Proposition 5.1. Let F £ £^ and z € E. Then zF=

DZF + D*ZF

in£^.

Let dt — Dht and (?t* = D\ , where ht = l[t,i]- Then we have Proposition 5.2. (i) For z eC and

/ Jo

z{t) dt
xeC.

and F £ ££,,

D*ZF=

[

z{t)d*tFdt,

Jo where the integral exists in the sense of Pettis integral.

172

The combination of Proposition 5.1 and Proposition 5.2 yields the following well-known relation: B(t) =

dt+dt.

Example 5.3 (Lee 1 5 ). Let

/ / y(s)z(s)ip(x)dsfi(dx). Jc Jo Symbolically, we have dsy= y(s). In particular,

Then we have Dzy = (y,z). dsB(t)= St(s). Similarly, for yi,... ,yn £ E*, we have

D z :y\---yn

:= ,n_1y

J2 (*Mi)> z">1 : ^ ( 2 ) ' " ' ^ ( " )

:

where •K rums through all permutations of { 1 , . . . , n}. Symbolically , we write ds : m • • -yn : =

(n

_ 1 ) , J2 y*d)(s)

:

y*V) • • '&(«) : •

In particular, da : B^)

• • • B(tn) : = _ 1 _ £

5K(1) (s) : B^2)

• • • BK(n)

:.

Example 5.4 (Lee 1 5 ' 2 0 ). Let T be a symmetric bounded linear operator on C. For (p £ £oo and z £ E, apply Example 4.8 and the integration by parts formula, one sees that ((Dz : (Tx, x)0 :,
f 2Tz(x)
^dx)

2Tx(s)z(s)ip(x)dsfi(dx).

Then ds :(Tx,x)0:=

2Tx(s).

In particular, when Tx(t) = JQ f(s) x(s) ds for / £ 1*2 [0,1], we have

: f f(u)B(u)) Jo

du: J = 2/(s)B(s).

173 Example 5.5 (Lee 1 5 ' 2 0 ). Forti,t2,t3,...,tn, d*tld^...dll=

we have :B(t1)B(t2)...B(tn):

Let / be twice Frechet differentiable at x in C, then the restriction of D2 f(x) to C is of trace class on C by Goodman's Theorem (see Kuo 12 ). Define the Laplacian A/(x) of / at x by A/(x) = T r ( D 2 / ( x ) | c ) - Then the number operator Nf(x) is defined by Nf(x) = — A/(x) + (x, Df(x)). Then N is a self-adjoint operator on £(*,. Moreover , we have Definition 5.6 (Lee 1 5 ). Let F G £^ and

)). Proposition 5.7 (Kuo 1 3 ,Lee 1 5 ). (i) For v? € 5oo, A

and A*F - J,,1

d;d;Fdt.

Transform

For a, /? e C and tp £ £ define the Fourier-Wiener transform J-a,p

Ta,p

:,

y € E*

(b) F(y) = -iD*y6,

y€E*

(c) T '• Btl • • • Btn := (—i)ndf • • • dtnSo, where So denotes the delta function concentrated at 0 £ E.

174

(d)

T : exp ( | jf B(t)2d?j

• Conditional

: =: exp ( i - jf B(t) 2 d/) :, (a ± 1)

expectation

Let £ „ = 0"({B(t,) : 0 = t0 < h < t2 • • • < tn = 1}). Theorem 5.9 (Huang-Lee 9 , Lee 2 0 ). Let

| Bn ] of ip relative to Bn enjoys the following integral representation

E[

It is easy to see that E[ip \ Bn ] £ SoTheorem 5.10 (Huang-Lee 9 ). Lett £ [0,1] and Bt be the a-field generated by B(u); 0 < u < t. Then, for (p G So, the conditional expectation E[<£> | Bt] of (p relative to Bt has an integral representation E[
J
Qty)n(dy)

for a.e. (/i) x in C, where Ptx(s) = x(s A t) for s £ [0,1] and Qt = I — PtMoreover, E[ tp \ Bt ] £ S0. Applying Theorem 5.9 and 5.10 we define the conditional expectation E[*|B„] and E[*|B t ] for a GWF $ as follows. Definition 5.11. For a GWF $ £ ££, we define «E[*|B n ], ^)>o = P , E[¥»|B„])>0; and «E[*|Bt],
6 The white noise integration • The Kubo-Takenaka formula A £ o - v a l u e d stochastic process X — {X(t) : t £ [0,1]} is called a generalized stochastic process. Definition 6.1. A generalized stochastic process {X(t) : t £ [0,1]} is called nonanticipating with respect to the filtration {B(t) : t £ [0,1]}, where B(t) — o~{B(u) : 0 < u < i], if for each t, X(t) is measurable with respect to B(t) in the sense that for all t £ [0,1], E[X(t)\B(t)} = X(t).

175 A generalized stochastic process {X(t) : t € [0,1]} is called a step process if there exist a partition {0 = to < t\ < • • • < tn = 1} such that X(t) = Xi for t £ [ti,ti+\). For a generalized nonanticipative step process X(t), we define the generalized stochastic integral with respect to Brownian motion in Ito sense by d

f

X{t)dB{i) := J2 Xi-iWU)

-Bfa-i)).

i=l

Jo

If X{t) is a nonanticipative continuous generalized process satisfying

I

^ \\X(t)\\*n

0

m

dt
for all m € N. We define

f X(t)dB(t) := lim £ *(«,_!)(£&) - 5(^-0) in S^, JO

ll^nll—0

^

where A n = {0 = t0 < t\ < • • • < tn = 1} is any partition of [0,1] and ||A n || is the mesh of A n . Theorem 6.2 (Huang-Lee 9 ). Let {X(t) : t £ [0,1]} be a nonanticipative continuous generalized process. If d

/ \\X(t)\\l.|2 Jo

n

dt < oo for all m € N,

then

f X(t)dB{t)= Jo

[ d*tX{t)dt Jo

Remark 6.3. When X(t) is further assumed to be satisfied the conditions: X(t)€£2(C,/i)and l

Jc Jo

2 \X(t\x)\2dtfi(dx)

< +oo,

the above theorem is exactly the well-known Kubo-Takenaka formula (see Kubo 6 and Kubo-Takenaka 7 ' 8 ). • The Clark formula for GWF's The representation of functional of Brownian motion by stochastic integral with respect to Brownian motion, known as the Clark formula, was first

176 studied in Clark 1 and later in Faria-Oliveira-Streit 2 and in Karatzas-OconeLi 11 . We shall reformulate of the Clark formula on the classical Wiener space for a class of GWF's. Note that even for rj G Ec, Pt(r))(s)(= r](s/\t)) lies in C* but not necessarily lies in Ec. Thus, we restrict our consideration only to the GWF's in £Q. Lemma 6.4 (Lee-Shih 2 2 ). For F G £% and r) G Ec, the mapping SF{Pt(rj)) is absolutely continuous as a function of one variable with respect tote [0,1]. For any TJ G EC, Lemma 6.4 implies that (d/dt) SF(Pt(r))) exists almost everywhere in (0,1) and (d/dt) SF(Pt(r])) is Lebesgue integrable so that SF(Pb(r,))-SF(Pa(r,))

= J

±SF(Pt(r,))dt,

0 < a < b < 1.

In particular, SF{r,) = ((F,l)) + £

±SF(Pt{r,))dt.

For any t G [0,1], let TF(t, •) be a complex-valued mapping on Ec given by TF(t,V) = jtSF(Pt(r,)),

r, G Ec.

Then Tp(t, •) is analytic on Ec. Applying Theorem 3.4, there exists a unique generalized function Kp(t) in ££, such that SKp(t)(r)) = Tp(t, rj) for each rj G Ec. The integral JQ Kp(t) dt exists in E^. Here, for a £^-valued function G on [0,1], the integral J 0 G(t) dt is defined as n

ttm V(fc-*<_!) G(0 in £^, provided that the limit exists, where A = {0 = to < ti < • • • < tn = 1} is any partition of [0,1], ||A|| is the mesh of A, and t^s are arbitrarily taken over [ti, U-i]. Moreover, for any ip G £oo,

[ KF(t)dt,
//oo

JO

Consequently, SF(ri) = ((F, 1)) + S ( f It follows that we have

KF(t) dt J (TI) for F G £0* and n G Ec.

177 Theorem 6.5 (Lee-Shih 2 2 ). Let F be in £Q. Then there exists a unique KF(t) £ SZo so that S(KF(t))(V)

=

±S(E[F\Ft]){V)

for any t £ [0,1] and r\ 6 Ec. Moreover, the integral f0 KF(t) dt exists in ££, and F=

((F,l)) + / Jo

KF(t)dt.

Let

(•) ^t G A f° r * e [0> !]• Then the conditional expectation E[Dtp(-) ht\!Ft] lies in L2(C, fi). By Theorem 6.2,

Jo

By a direct computation, the equality (6.1) becomes / T,{t)S(p[D
f Jo

f,(t)S(D
= £ jtS
Jo

a.e. (fi) x on C,

where the integral is the Wiener-Ito stochastic integral. • Generalized ltd formula for Gaussian Processes For / £ S, the Schwartz space on R, and a nonzero h S C , jf f(h(x)) f(y) dy^j
f°°

V27T J-oo

e-Wy2W2of(y)Sp(iyh)dy,

178

where f is the Fourier transform of / . Since the mapping y i-» -(i/2)y IMo S
tpeSvo,

(6.2)

where Gh,v(y) = l / v ^ e - * 1 / 2 ^ ! ? (Sip)(iyh). Then F(h) € £^. Next consider the Gaussian process Z(t) represented by Zt(x) = (x,zt) with zt € C. Assume that the L2P, l]-valued function zt is differentiable with derivative kt, i.e. lim

Zt+e

~ ** = * t in X2[0,1] ,

(6.3)

We shall derive the Ito formula for F(Z(t)) for any tempered distribution F on R. By (6.2), for 0 < t < 1, F(Z(t)) is defined as a generalized function in ^oby

pxz(i)), € £oo. Differentiate ((F(Z(t)),
F(Z(b)) = F(Z(a)) + J" D*kt F'(Z(t))dt + i jf r(t)F"(Z(t)) dt,

(6.4)

where 0 < a < b < 1. Remark 6.8. The term Ja D£ F'(Z(t))dt tegral. For example,

is interpreted as a stochastic in-

(a) If Z(t) is the Brownian motion, then zt(s) = s At , r(t) = t and kt(s) = l[ t l ](s). In this case, we have rb

rb

I Dlt F'(Z(t))dt = J F'(Z(t)dZ(t). Ja

Ja

(b) If Z(t) is the Brownian bridge, then zt(s) = s A t - st , r(t) = t(l — t) and kt(s) = l[t,i](«) — s- I n this case f D*kt F'(Z(t))dt Ja

= f Ja

F'(Z(t)dZ(t+)

+ f tf"(X{t))dt Ja

in S(L2)*

179 for 0 < a < b < 1, where the second term represents the forward integral denned as follows: Let {Yt : a < t < b},0 < a < b < 1 be a. continuous £ o - v a m e d process. We define /

YtdX(t+):=

Urn f > * ; ~ ^ - J ^

in 5 ^

if this limit exists, where r = {a — to < t\ < • • • < tn = b} (see Kuo 13 ). Using the forward integral, then, for and F € C 2 (R), Ito formula becomes F{Z{b))=

F(Z(a)) + j " F'(Z(t))dZ(t+)

+ \ j " F"(Z(t))dt.

(6.5)

(c) If Z(t) is a semimartingale version of the Brownian bridge represented by Z(t,x) = (x,zt) with J3t(s) = ( t - l ) l n ( l - s A t ) a n d ^ ( s ) = 0, 0 < s < 1. Then r(t) = t(l - t) and kt = l[t,i](s) + ln(l - s At) for s € [0,1], and for 0 < t < 1. In this case the equation (6.5) holds with

/ F'(Z(t))dZ(t+)= J F'(Z(t))dZ(t). Ja

Ja

References 1. J. M. C. Clark: The representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Stat. 41 (1970), 1281-1295; 42 (1971), 1778 2. M. de Faria, M. J. Oliveira, and L. Streit: A generalized Clark-Ocone formula, Preprint, 1999. 3. L. Gross: Abstract Wiener space, In "Proceedings 5th Berkerley Symp. Math. Stat. Prob." Vol.2 (1965), 31-42 4. T. Hida: Analysis of Brownian Functionals, Carleton Mathematical Lecture Notes 13, 1975 5. T. Hida, H.-H. Kuo, J. Pottohoff, and L. Streit: White Noise: An Infinite Dimensional Calculus, Kluwer Academic Pubblishers, 1993 6. I. Kubo: Ito formula for generalized Brownian functionals, Lecture Notes in Control and Information Sciences 49, 156-166, Springer-Verlag, 1983 7. I. Kubo and S. Takenaka: Calculus on Gaussian white noise III, Proc. Japan Acad. 57A (1981), 433-437 8. I. Kubo and S. Takenaka: Calculus on Gaussian white noise IV, Proc. Japan Acad. 58A(1982), 186-18

180

9. H.-C. Huang and Y.-J. Lee: Conditional expectation of generalized Wiener functionals, 2001, Preprint 10. G. Kothe: Topological Vector Space I, Springer-Verlag, New York/ Heidelberg/ Berlin, 1966 11. I. Karatzas, D. Ocone, and J. Li: An extension of Clark's formula, Stochastic and Stochastic Reports 37 (1991), 127-131 12. H.-H. Kuo: Gaussian Measures in Banach Spaces, Lectures Notes in Math. Vol. 463, 1975 13. H.-H. Kuo: White Noise Distribution Theory, CRC Press, 1996 14. Y.-J. Lee: Sharp inequalities and regularity of heat semigroup on infinite dimensional space, J. Fund. Anal. 71 (1987), 69-87 15. Y.-J. Lee: Generalized functions on infinite dimensional spaces and its application to white noise calculus, J. Fund. Anal. 82 (1989), 429-464 16. Y.-J. Lee: A reformulation of white noise calculus, in "White Noise Analysis- Mathematics and Applications", Edited by T. Hida, H.H. Kuo, J. Potthoff and L. Streit, 274-293, World Scientific (1990). 17. Y.-J. Lee: Analytic version of test functionals, Fourier transform and a characterization of measures in white noise calculus, J. Fund. Anal. 100 (1991), 359-380 18. Y.-J. Lee: Positive generalized functions on infinite dimensional spaces, In " Stochastic Process, a Festschrift in Honour of Gopinath Kallianpur", Springer-Verlag, 1993, 225-234 19. Y.-J. Lee: Integral representation of second quantization and its application to white noise analysis, J. Fund. Anal. 133 (1995), 253-276 20. Y.-J. Lee: Generalized white noise functionals on classical Wiener space, J. Korean Math. Soc. 35 No.3 (1998), 613-635 21. Y.-J. Lee and C.-C. Huang: Ito formula for functional of Brownian bridge, preprint 22. Y.-J. Lee and H.-H. Shih: The Clark formula of generalized Wiener functionals, In "Quantum Information IV", to appear. 23. P. Malliavin: Stochastic calculus of variation and hypoelliptic operators, Proc. Int. Sym. S.D.E. Kyto, 1976, Kinokuniya,1978, 195-263 24. P. A. Meyer and J.-A. Yan: Les "fonctions caracteeristiques" des distribution sur l'espace de Wiener, Sem. Probab. XXV, Lecture Notes in Math. 1485 (1991), 61-78 25. D. Ocone: Malliavin's calculus and stochastic integral representations of functionals of diffusion process, Stochastics. 12 (1984), 161-185 26. S. Watanabe: Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Inst, of Fundamental Research, Bombay, 1984

Quantum Information V Eds. T. Hida and K. Saito (pp. 181-191) © 2006 World Scientific Publishing Co.

F R A C T I O N A L B R O W N I A N M O T I O N S A N D T H E LEVY LAPLACIAN *

KENJIRO NISHI, KIMIAKI SAITO Department

E-mails:

of Information Sciences Meijo University Tempaku, Nagoya 468-8502, Japan [email protected]; [email protected] ALL A N U S H. T S O I Department of Mathematics University of Missouri Columbia, MO 65211, U.S.A. E-mail:[email protected]

In this paper we give a relationship between fractional Brownian motions and the Levy Laplacian acting on a space of white noise distributions introducing an operator changing the white noise B{t) by e iB (*>. In addition, we give a stochastic expression of the semigroup generated by the powers of the Levy Laplacian through the operator.

1. Introduction The Levy Laplacian was introduced by P. Levy 22 . Hida 7 discussed this Laplacian in his theory of white noise functionals. Accardi, Gibilisco, Volovich 2 obtained an important relationship between this Laplacian and the Yang-Mills equations. It has been studied by many authors ( see Refs. 1,3-6,8,12,19,20,21,23-25,28-30,32-38, etc). In the previous paper 36 we extended the Levy Laplacian to a self-adjoint operator densely defined on a Hilbert space and also gave a relationship between the Laplacian and an infinite dimensional Ornstein-Uhlenbeck process through an operator changing the white noise B{t) by elB^\ "This work is supported by the Joint Research Project "Quantum Information Theoretical Approach to Life Science" for the Academic Frontier in Science promoted by the Ministry of Education in Japan.

182

The purpose of this paper is to give a relationship between the fractional Brownian motions and the Levy Laplacian acting on white noise distributions, through an infinite dimensional fractional Brownian motion. We also give a stochastic expression of the semigroup generated by the power of the Levy Laplacian. The paper is organized as follows. In Section 2 we summarize some basic definitions and results in the white noise analysis. In Section 3, following our previous paper 3 6 , we explain a self-adjointness of the Levy Laplacian and give an equi-continuous semigroup of class (Co) generated by the powers of the Levy Laplacian. In Section 4, following our previous papers 34>35, we explain a relationship between the Levy Laplacian and the number operator introducing an operator changing the white noise B(t) by elB^\ In the last section we give an infinite dimensional fractional Brownian motion and a relationship to the Levy Laplacian. A stochastic expression of a semigroup generated by the power of the Levy Laplacian is also obtained through the operator in Section 4. 2. Preliminaries In this section we assemble some basic notations of white noise analysis following Refs. 10, 16, 19 and 26. We take the space E* = <S'(R) of tempered distributions with the standard Gaussian measure fj, such that J

exp{t(x,0} dti(x) = exp ( - ^ | § ) ,

£ € £ = 5(R),

where | • |o is the L 2 (R)-norm, and (•,•) is the canonical bilinear form on E* x E. Let A — — (d/du)2+u2+l. This is a densely defined self-adjoint operator 2 on L (R) and there exists an orthonormal basis {e„; v > 0} for L 2 (R) such that Aev = 2(i/ + l)e„. We define the norm | • \p by | / | p = \Apf\0 for / G E and p G R, and let Ep be the completion of E with respect to the norm | • \p. Then Ep ia a real separable Hilbert space with the norm | • | p and the dual space E'p of Ep is the same as £ L P (see Ref. 16). Let E be the projective limit space of {Ep;p > 0} and E* the dual space of E. Then E becomes a nuclear space with the Gel'fand triple E C L2(R) C E*. We denote the complexifications of L 2 (R), E and Ep by L ^ R ) , Ec and Ec,P, respectively.

183 The space (L 2 ) = L2(E*,/j.) of complex-valued square-integrable functional denned on E* admits the well-known Wiener-Ito decomposition: oo

(i 2 ) = 0ffn, 71=0

where Hn is the space of multiple Wiener integrals of order n € N and Ho = C. Let Lc(R)® n denote the n-fold symmetric tensor product of L2C(R). If ip G (L2) is represented by

Mlo= £n!|/„|g =0

where | • |o means also the norm of L2-;(R)lS>™. For p G R, let ||<£>||p = ||r(A) p (^||o, where T(A) is the second quantization operator of A. If p > 0, let (E)p be the domain of Y(A)P. If p < 0, let (Z?)p be the completion of (L 2 ) with respect to the norm || • || p . Then (E)p, p G R, is a Hilbert space with the norm || • || p . It is easy to see that for p > 0, the dual space (E)p of (E)p is given by [E)_p. Moreover, for any p G R, we have the decomposition oo

(£)„ = ©*#>, n=0

where # 4 is the completion of { I n ( / ) ; / G £ § " } with respect to || • || p . Here E®n is the n-fold symmetric tensor product of EQ. We also have Hn = {In(/); / G E^p} for any p G R, where £ § ^ is also the n-fold symmetric tensor product of Ec,p- The norm || = £^°=o !«(/«) € ( £ ) p is given by / oo

\ 1/2

IMIp=(£nl|/»l2J

- /»€£§£,

where the norm of i?®" is denoted also by | • | p . The projective limit space (E) of spaces (E)p, p G R is a nuclear space. The inductive limit space (E)* of spaces (E)p,p G R is nothing but the dual space of (E). The space (E)* is called the space of generalized white noise junctionals. We denote by the canonical bilinear form on

184

(E)* x (E). Then we have OO

n=0

for any $ = £ ~ = 0 I n ( F n ) G (£)* and ^ - £ ~ = 0 I n ( / » ) G (E), where the canonical bilinear form on (E®n)* x ( £ ^ n ) is denoted also by (•,•). Since exp(-,£) G (E), the S-transform is defined on (£)* by 5 [ * ] ( 0 = exp ( - | « , 0 ) « * , e x p ( - , 0 » ,

£ G £C-

3. Self-adjointness of the Levy Laplacian The .S-transform F G S[(E)*] has a property that for any £,77 G i?c the function F(£ + zr\) is an entire function of z G C. Hence we have the series expansion: 00

n

n=0

where F^™) (£) : EQ, X • • • X F C —> C is a continuous n-linear functional. Fix a finite interval T in R. Take an orthonormal basis {C„}$£L0 C E for L {T) satisfying the equally dense and uniform boundedness property (see Refs. 7,10,9,20,22,25,28 and 30). Let VL denote the set of all $ G (E)* such that the limit 2

JV-l

*LS[$](.Q = ^

Jf £ W ( 0 ( C n , Cn) n=0

exists for any £ G F c and is in S[(E)*\. The Levy Laplacian A/, is denned by AL$ = 5_1AL5$ for $ € X>L- We denote the set of all functional $ G T>i such that 5[$](?7) — 0 for all 77 G F with suppfa) C T c by 2?J. Take a white noise distribution $ whose S-transform S[$](£) is given by S[$](0= /

/ ( u ) e i a i l ( u i ) . . . eia^{u^du,

(3.1)

185 / e £ § n , o t € R , k = l , 2 , . . . ,n. Using the Wick ordering : by /

: we write the generalized white noise functional

/ ( « i , . . . ,un):eiaiX^Ul)---eia"x^)

: du,

(3.2)

This functional is in 2?£ and an eigenfunction of the operator A L as follows. Theorem 3.1. 36 A generalized white noise functional $ as in (3.2) satisfies the equation AL* = ~ X > 2 * .

(3.3)

Put D n = J J ^ / ( u ) : f[ ete("-> : du € X>£; / e Ljj(R) & n n L^(R)® n 1 for each n £ N U {0}. Then D n is a linear subspace of (E)-p for any p > Y5, (see 21 ) and A L is a linear operator from D„ into itself such that | | A i $ | | _ p = Tyj||$||_ p for any $ € D n . Define a space D„ by the completion of D n in (E)-p with respect to || • ||_ p . Then for each n € NU{0}, D n becomes a Hilbert space with the inner product of (E)_p. For each n S N U {0}, the operator A^ becomes a continuous linear operator A/, from D n into itself satisfying | | A L $ | | _ P = — | | $ | | _ p for any $ G D „ .

N

(

\

2 £

Put a^v (n) = ^2t=0 (i^i)

{

oo

and define a space

E_ P ) JV

by

oo

5^*Be(E)*;5^aAr(n)||$B||lp
n=0

with the norm j | | • |||- P ,JV given by

(

oo

\ V2

^a N (n)||$ n || 2 _ p n=0

oo

, $ = 53$ n GE_ P l J v /

n=0

186 for each N e N U {0} and p > ^ . Then for any N e N and p > ^ , E_ P i i V is in ( £ ) - ? and is a Hilbert space with respect to the norm ||| • |||_ p ,;vWith the projective limit space E_ P i 0 0 = Dw=i E-P,N, ing inclusion relations:

we have the follow-

E_p,oo C • • • C E_ P i i V + i C E_p,iv C • • • C E _ P i l c ( £ ) - P . For any n € NU {0}, D„ is included in E_p,oo. The operator Az, can be extended to a continuous linear operator defined on E_ Pi 2 into E _ P J I , denoted by the same notation AL, satisfying |||A£$|]|_ P) ./v < |||$|||_ p ,./v +1 , N = 1,2,3,... , $ € E_ P i 0 0 . With these properties, we have the following: Theorem 3.2. 36 The operator A^ becomes a self-adjoint operator densely defined on E_ Pi ;v for each N > 1 and p > j%. Let 7 > 0. Define a linear operator G] on E_ P i 0 0 by oo n=0

for each t > 0 and $ = £ £ L 0 $ „ € E_p,oo, $ n e D „ , n = 0 , l , 2 P r o p o s i t i o n 3 . 1 . 35 Let 7 > 0 a n d p > ^ . Then {GJ;t > 0} is an equicontinuous semigroup of class (Co) generated by - ( - A / , ) 7 as a continuous linear operator from E_p,oo into itself.

4. A relationship between the Levy Laplacian and t h e number operator Put

{

00

00

$ = £ l „ ( / „ ) € (£)_,; S[*](e*) = £ > * ) ® n , / n ) n=0

"=0

exists in 5[E_ P and define an operator K on £_ p by if[$]=S-1[5[$](e«)].

187 The operator K implies a relationship between AL and the number operator Jf on (E)* given by oo

tf$ = £ n I „ ( / „ ) for 4 = £ ~ = o M / » ) 6 ( £ ) * , n=0

as follows.

Proposition 4.1.

36

For any $ € £_ p we ftai/e

ALK[S] = -±KM*]]Put

{

OO

OO

n=0

n=0

s u p p ( / n ) c T,n = 0 , 1 , 2 , . . . } for g > 0 and N G N . Define a space [£^]q,jv by the completion of [E]g^ with respect to the norm || • ||T™— given by / oo

\ 1/2

for 0. Put = C\q>o [E]q,N with the projective limit topology and also put [£]co,oo = rW>i [^]OO,JV with the projective limit topology. The following fact is proved in Ref. 35. [•EJOO.JV

Proposition 4.2. 3 5 Let p > ^ . Then the operator K is a continuous linear operator from [-E]<x>,oo into E_ P i 0 0 .

188 5. An infinite dimensional fractional Brownian motion and the Levy Laplacian Let {B2(t); t > 0}, k = 0 , 1 , 2 , . . . , be an independent sequence consisting of fractional Brownian motions with the parameter 0 < 7 < 2, which has the covariance function:

The operator K implies a relationship between the semi-group {G~l; t > 0} and an infinite dimensional stochastic process {U^\t > 0} starting at x e E* given by

u x)=exp

^

x+

exp

dB 7 (s)

{-w\){ ^\Io (w\

, t > 0,

where {B 7 (t);t > 0} is an infinite dimensional fractional Brownian motion defined by 00

k=0

Then we have the following: Theorem 5.1. For any t > 0 and

(5-1)

Proof. For ip = exp{(-,r?) — \{r),rf)}, rj G Ec we can check that e-&"
= E[
x€E*.

Since a linear span of exp{(-,77) — \{r),rj)}, 77 € Ec is dense in (E) and [E]ca,oo is in (E), we obtain (5.1). • Let fJ^(x, A) be a transition function given by

lil(x,A) =e-t(WY\A{x),

xeE*,

t > 0,

where A is a bounded Borel set. We can define the function n1(-,A) in (L2) for each t > 0 and bounded Borel set A. Then we have the following: Theorem 5.2. For any t > 0 and

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