Table Of ContentSelected 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
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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, 1927- II. Accardi, L. (Luigi), 1947-
QA274.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.
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ProfTakeyukiHida
Vll
Preface
Takeyuki Hida is a visionary thinker. In 1975 he introduced white noise the
ory 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 phe
nomena 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)]b - f* f(t)B(t)dt
a
for a C1-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 J B(l) dB(t) is
0
not defined as an Ito integral. Hida envisioned that this integral should be viewed
directly as J B(l)B(t) dt, namely, as a white noise integral.
Q
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, A33,
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 oper
ator. 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(£) = / p(t)£(t)2 dt defined on L2(0,1). This function can be regarded
0
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 correspond
ing infinite dimensional Laplacian of F should be AF(£) = 2 J p(t) dt. Hida was
Q
inspired by Levy's idea along this line and tried to interpret nonlinear functions
defined on L2(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) = lim_).o e~1(B(t + e) — B(t)) does not exist
e
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.^o[6(t, e) — e_1] exists with respect to the weaker norm. This renormalization
e
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 L2(R). However, for the white noise
functional B(t)2, the function / is S ® S which is not in L2(R). Thus Hida's idea
t t
to define :B{t)2: is a "singular" renormalization and thus :B(t)2: can be regarded
as the "generalized" Wiener integral of 5 (gi5. The same idea is carried out for each
t t
n and then take the orthogonal direct sum of the new spaces over n to get a space
(L2)~ 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(£) = J. e^x'^<f(x)dfj,(x) in his Carleton Uni
s
versity lecture notes as a fundamental tool to identify Brownian functionals. This
transform has its root in the T-transform of his 1970 book. Under this transform,
the white noise B(t) being regarded as a generalized Brownian functional corre
sponds to the function i^(i)exp[-| f £(u)2 du]. The T-transform is Hida's vision
R
to interpret Levy's functional analysis. A related 5-transform given by Sip(£) =
J. ip(x + £) dfj,(x) was later used by Kubo and Takenaka in 1980. For example, the
s
function F(£) = JQ p{t)£{t)2dt mentioned above in Levy's work is the S-transform
IX
of the generalized Brownian functional J p(t): B(t)2 : dt. The S-transform, when
0
restricted to the space L2(S*,fi), had already been introduced by Segal and
Bargman before 1975. However, it was Hida's ingenious idea to use the T-transform
(equivalently the .^-transform) as a tool to define and study generalized Brownian
functionals.
White noise has long been regarded as an informal, mysterious, and imaginary
object. Owing to the tremendous effort of Hida, there is a mathematical theory of
white noise. We see many applications of this theory to Feynman integrals, Dirichlet
forms, quantum probability, random fields, calculus of variation, infinite dimensional
harmonic analysis, infinite dimensional rotation groups, stochastic partial differen
tial equations, biology, to name just a few. And now, 25 years since 1975, white
noise theory is listed as 60H40 under stochastic analysis in the MSC 2000 subject
classifications.
Hida has written five books (two of them originally in Japanese), translated two
books, and published more than one hundred papers. This volume contains his
Carleton University lecture notes and 30 selected papers. The Carleton University
lecture notes have been retyped to correct some misprints, but the page numbers
are preserved. The papers in this volume can be divided into several categories:
1. General theory of white noise functionals: [l]-[7]
2. Gaussian and other processes: [8]-[10]
3. Infinite dimensional harmonic analysis rotation group: [11]-[18]
4. Quantum theory: [19]-[21]
5. Feynman integrals and Dirichlet forms: [22]-[25]
6. Variational calculus and random fields: [26]-[30]
7. Application to biology: [31]
At the end of this volume there are comments on these selected papers. We hope
that they will give the reader some guidelines to understand Hida's ideas.
From "My mathematical journey" (pp. 453-473) we see how Hida decided to
study mathematics, how he was influenced by P. Levy and other mathematicians
and his vision of mathematical science. In recent years Hida has developed the habit
of looking everything from the white noise viewpoint. He uses the term "idealized
elemental random variables" to express the system {B(t)\t G R}. In his mind, these
i.e.r.v. are the most important mathematical elements to study natural phenomena
involving fluctuations.
We want to thank Professors M. Hitsuda and I. Kubo for writing comments on
some of Hida's papers. Thanks also go to Professor L. Pitt for allowing us to use
his remark on Hida's Lecture in Pittsburgh in 1975.
Finally we would like to thank World Scientific for publishing this volume.
The editors
