Selected Papers of
Takeyuki Hida
Selected Papers of
Takeyuki Hida Edited by
L Accord i Universita di Roma Tor Verga...
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Selected Papers of
Takeyuki Hida
Selected Papers of
Takeyuki Hida Edited by
L Accord i Universita di Roma Tor Vergata,
Italy
HHKuo Louisiana State University,
USA
N Obata Nagoya
University,
Japan
K Saito Meijo University,
Japan
Si Si Aichi Prefectural University,
Japan
L Streit University of Bielefeld, Germany and University of Madeira, Portugal
fe World Scientific \m
Singapore • New Jersey • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data Selected papers of Takeyuki Hida / edited by L. Accardi. . . [et al.]. p. cm. Includes bibliographical references. ISBN 9810243332 (alk. paper) 1. Stochastic processes. I. Hida, Takeyuki, 1927II. Accardi, L. (Luigi), 1947QA274.H5313 2000 519.2-dc21
00-063295
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
The editors and publisher would like to thank the following organisations and publishers of the various journals and books for their assistance and permission to reproduce the selected reprints found in this volume: Academic Press Academie des Sciences, Institut de France Circolo Matematico de Palermo Elsevier The Japan Academy Kyoto University Press Nagoya University Press Nanka Publishers Russian Academy of Science Springer-Verlag University of California Press While every effort has been made to contact the publishers of reprinted papers prior to publication, we have not been successful in some cases. Where we could not contact the publishers, we have acknowledged the source of the material. Proper credit will be accorded to these publishers in future editions of this work after permission is granted.
Copyright © 2001 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.
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Printed in Singapore.
Prof Takeyuki Hida
Vll
Preface
Takeyuki Hida is a visionary thinker. In 1975 he introduced white noise theory in his monograph "Analysis of Brownian Functionals" (Carleton Mathematical Lecture Notes No. 13) which has generated a tremendous amount of research during the last 25 years. His ideas and work are truly influential in many ways. White noise is referred to as a sound with equal intensity at all frequencies within a broad band. The word "white" is used because of its similarity to "white light" which is made up of all different colors (frequencies) of light combined together. In applied science white noise is often taken as a mathematical idealization of phenomena involving sudden and extremely large fluctuations. It is informally denned as a stochastic process z(t) such that the z(t)'s are independent and for each t, z(t) has mean 0 and variance oo in the sense that E(z(t)z(s)) = S(t — s), where S is the Dirac delta function. Obviously z(t) is not an ordinary random variable for each t. White noise can also be regarded as the derivative B{t) of a Brownian motion B(t). Since \B(t + h) - B(t)\ ~ \h\x^2 for small h, B(t) does not exist in the ordinary sense for each t. The white noise B(t) had long been used in integration before 1975 by engineers like A. V. Balakrishnan. For example, the integration by parts formula can be used to define the informal integral f*f(t)B(t)dt as f{t)B{t)]ba - f* f(t)B(t)dt 1 for a C -function / . On the other hand, in Ito's theory of stochastic integration B{t) is combined with dt to define the integral J f(t)B(t) dt as the Ito integral / f(t) dB(t) for a non-anticipating stochastic process /(£) with almost all sample paths being square integrable. However, a simple integral such as J0 B(l) dB(t) is not defined as an Ito integral. Hida envisioned that this integral should be viewed directly as JQ B(l)B(t) dt, namely, as a white noise integral. A few years before 1975 Hida had advocated his ideas of white noise theory in various conferences. What he had in mind was to introduce a mathematical theory so that B(t) is meaningful for each t and the collection {B(t)\t G K} can be used as a continuum coordinate system. Moreover, he outlined his vision of complex white noise analysis, random fields, infinite dimensional rotation groups, infinite dimensional harmonic analysis, etc. Looking back more than 25 years ago, one cannot help wondering how Hida came up with such a bold attempt. Perhaps it is best described by a remark of Loren Pitt referring to Hida's lecture for the 1975 Multivariate Analysis conference in Pittsburgh "I did not understand all the mathematics in his lecture, but I have the feeling that it is something going to be very important." It is true that Hida was influenced by Paul Levy's work on Brownian motion and functional analysis. But his mathematical theory of white noise also stemmed from his own earlier work, in particular, the paper "Canonical representation of Gaussian processes and their applications (Memoirs Coll. Sci., Univ. Kyoto, A 3 3 ,
vm 1960, 109-155) and the book "Stationary Stochastic Processes" (Princeton Univ. Press, 1970). In the canonical representation of Gaussian processes, known as the Cramer-Hida theorem, we can see the root of the white noise differentiation operator. In his latter book we find Hida's vision of harmonic analysis on white noise space which eventually led to the concepts of generalized multiple Wiener integrals and generalized Brownian functionals in his Carleton University lecture notes. Paul Levy discussed harmonic analysis on the space L2(0,1) in his book "Lecons d'analyse Fonctionnelle" (Gauthier-Villars, 1922). Consider, for example, a simple function F(£) = / 0 p(t)£(t)2 dt defined on L 2 (0,1). This function can be regarded as an infinite dimensional analogue of the function f(x) = ^fc=i akxj. on R™. The Laplacian of / is given by Af(x) = 2 £]/b=i ak, which suggests that the corresponding infinite dimensional Laplacian of F should be AF(£) = 2 JQ p(t) dt. Hida was inspired by Levy's idea along this line and tried to interpret nonlinear functions defined on L 2 (0,1) and the Levy Laplacian of such functions from the white noise viewpoint. Here is Hida's idea to define B(t) for each t and nonlinear functions of {B(t)\t £ R} as generalized Brownian functionals. Let p, be the standard Gaussian measure on the dual space S* of the Schwartz space S on the real line R. The probability space (<S*, p) is called a white noise space since its elements can be regarded as informal sample paths of white noise. Apply the Wiener-Ito theorem to decompose L2(S*, p) into an orthogonal direct sum L2(S*,p) = X ^ o ^ « °^ multiple Wiener integrals or homogeneous chaos. The limit B(t) = lime_).o e~1(B(t + e) — B(t)) does not exist in the space "W\. A weaker norm on Hi is introduced so that the limit exists. Next consider the function B(t)2, which is the limit of 6(t,e) = (e _ 1 [B(i + e) - B(t)})2 as e -> 0. Obviously this limit does not exist in any sense since the expectation of 9(t, e) is e _ 1 , which has no limit as e —>• 0. In order to get a well-defined quantity out of B(t)2, consider 6(t, e) — e _ 1 and take the limit. But the limit does not exist in the space %%• A weaker norm on H2 is introduced so that the limit : B(t)2 := \im.e^o[6(t, e) — e _1 ] exists with respect to the weaker norm. This renormalization procedure was in fact already used by K. Ito in 1951 to define multiple Wiener integrals, namely, the multiple Wiener integral of /® 2 is given by (Ii(f))2 - \\f\\2, where h(f) is the Wiener integral of / G L 2 (R). However, for the white noise functional B(t)2, the function / is St ® St which is not in L 2 (R). Thus Hida's idea to define :B{t)2: is a "singular" renormalization and thus :B(t)2: can be regarded as the "generalized" Wiener integral of 5 t (gi5t. The same idea is carried out for each n and then take the orthogonal direct sum of the new spaces over n to get a space (L 2 )~ of generalized Brownian functionals. Let {L2)+ be the corresponding space of test functionals. The resulting triple (L2)+ C L2(S*,n) c (L2)~ is an infinite dimensional analogue of the Gel'fand triple S(Rd) C L2(Rd) C S*{Rd). Hida used the T-transform 7V(£) = Js. e^x'^ € Cl(P), denote the standard Brownian motion, namely {B(t)} is a Gaussian process with E(B(t)) = 0 and E{B(t)B(s)} = i(|*| + \s\ — \t — s|). We shall consider functionals such as
f(B(t);teT) and discuss their analysis. They are simply called Brownian functionals. In order to discuss the analysis we would like to express them in the form 0,3 = 1,2,...} o/ P ^ a^o dense in (L2). Since any £ in E is expressed as ( = ^
$> = »•
15 From the definition of T we have (T] I I ^
(~f^)
^(a;)
(using independence)
= exp
-jiia'n^s^p
•^II^-II 2
(this equality will be established later) : exp
-W (v^os^n^'O'
where -F(i 1 ,..., tn) is the symmetrization of ^j1® ® £22® ® • • •. We claim that (15)
(Tip)® e Tn.
In fact, starting with Cn(-,rj) = C{-)C(jiy ,ri\ , r\ £ S, we have more general expression in the case rj = X^fc^fc- Then, if the limit is also taken into account, we immediately prove (15). To establish the formula (16)
/exp[ia(x, V )]H k
(^S)
dfi(x) = (V2i)kak exp
- y
Nl = i,
we recall the generating function of Hermite polynomials exp[2te — t2]. Set