Table Of ContentW hite Noise
Distribution
Theory
W hite Noise
Distribution
T heory
Hui-Hsiung Kuo
CRC Press
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Preface
White noise is generally regarded as the time derivative of a Brownian
motion. It does not exist in the ordinary sense since almost all Brownian
sample paths are nowhere differentiable. Informally it can be regarded as a
stochastic process which is independent at different times and is identically
distributed with zero mean and infinite variance. Due to this distinct fea
ture, white noise is often used as an idealization of a random noise which
is independent at different times and has large fluctuation. This leads to
integrals with respect to white noise. Such integrals can not be defined
as Riemann-Stieltjes integrals. In order to overcome this difficulty, K. Ito
developed stochastic integration with respect to Brownian motion in 1944.
The Ito theory of stochastic integration turns out to be one of the most
fruitful branches of mathematics with far-reaching applications.
In the Ito theory of stochastic integration, the integrand needs to be
nonanticipating. It is natural to ask whether it is possible to extend the
stochastic integral wherein the integrand can be anticipating. In the 1970s
such integrals were defined by M. Hitsuda, K. Ito, and A. V. Skorokhod.
Since then there has been a growing interest in studying such integrals and
anticipating stochastic integral equations. There are several approaches to
define these integrals; for instance, the enlargement of filtrations, Malliavin
calculus, and white noise analysis. An intuitive approach is to define such
an integral directly with white noise as part of the integrand. This would
require the building of a rigorous mathematical theory of white noise.
Motivated by P. Lévy’s work on functional analysis, T. Hida introduced
the theory of white noise in 1975. The idea is to realize nonlinear functions
on a Hilbert space as functions of white noise. There are several advantages
for this realization. First of all, white noise can be thought of as a coordinate
system since it is an analogue of independent identically distributed random
variables. This enables us to generalize finite dimensional results in an
intuitive manner. Secondly, we can use the Wiener-Ito theorem to study
functions of white noise. This allows us to define generalized functions of
white noise in a very natural way. Thirdly, since Brownian motion is an
integral of white noise, the use of white noise provides an intrinsic method
to define stochastic integrals without the nonanticipating assumption.
During the last two decades the theory of white noise has evolved into an
Preface
infinite dimensional distribution theory. Its space (£) of test functions is
the infinite dimensional analogue of the Schwartz space 5(R'’) on the finite
dimensional space R'*. The dual space (£)* of (5) is the infinite dimensional
analogue of the space 5'(R'') of tempered distributions on R'’. It contains
generalized functions such as the white noise B{t) for each fixed t. Thus
B(t) is a rigorous mathematical object and can be used for stochastic inte
gration. White noise distribution theory has now been applied to stochastic
integration, stochastic partial differential equations, stochastic variational
equations, infinite dimensional harmonic analysis, Dirichlet forms, quan
tum field theory, Feynman integrals, infinite dimensional rotation groups,
and quantum probability.
There have been two books on white noise distribution theory:
1. T. Hida, H. -H. Kuo, J. Potthoff, and L. Streit: White Noise: An Infinite
Dimensional Calculus, Kluwer Academic Publishers, 1993.
2. N. Obata: White Noise Calculus and Fock Space. Lecture Notes in
Math., Vol. 1577, Springer-Verlag, 1994.
The space (£*)* of generalized functions used in these books is somewhat
limited. By the Potthoff-Streit characterization theorem, the 5-transform
5$ of a generalized function $ in {£)* must satisfy the growth condition
|5$(0I <Kexp [a|i|2].
Thus the function F{^) = exp [(^, is not the 5-transform of a general
ized function in (S)*. However this function is within the scope of Lévy’s
functional analysis. In fact, F(^) = exp is the 5-transform of a
solution of the heat equation associated with the operator (A^)^. Here A^
is the adjoint of the Gross Laplacian A^. This shows that there is a real
need to find a larger space of generalized functions than
Recently, Yu. G. Kondratiev and L. Streit have introduced an increasing
family of spaces (5)^, 0 < ^ < 1, of generalized functions with (S)¡ = (£)*•
In the characterization theorem for $ in (£)’^, the growth condition is
|5$(0l<i^exp[a|e|‘^ ].
For example, the above function F(^) = exp 5-transform of
a generalized function in
There are three objectives for writing this book. The first one is to
carry out white noise distribution theory for the space (5)^ of generalized
functions and the space (S)^ of test functions for each 0 < /? < 1. As
pointed out above, the space (S)* is not large enough for the study of
heat equation associated with the operator In general, F(^) =
exp [f (^, is the 5-transform of a solution of the heat equation associated
with the operator (A^)*'. The solution is a generalized function in the space
Preface
i^){k-i)/k' There are other motivations for the spaces 0 < /? < 1; for
instance, a generalization of Donsker’s delta function in Example 7.7 and
the white noise integral equation in Example 13.46. An important example
of generalized functions in the space is given by the grey noise measure
u\ (see Example 8.5) which has the characteristic function
f
dux{x) = I-A(klo)»
Je»
where La, 0 < A < 1, is the Mittag-Leffler fimction given by
S r ( l + An)-
Here T is the Gamma function. The grey noise measure induces a general
ized function in the space (£^)Î_a*
The second objective is to discuss some of the recent progress on Fourier
transform, Laplacian operators, and white noise integration. The Fourier-
Gauss transforms acting on the space {S)^ of test functions are studied in
detail. The probabilistic interpretation of the Lévy Laplacian is discussed.
The Hitsuda-Skorokhod integral d*(p{8)d8 is defined as a random vari
able, not as a generalized function. The 5-transform is used to solve some
anticipating stochastic integral equations. For instance, consider the antic
ipating stochastic integral equation in Example 13.30
X(i) = sgn(B(l))+ fd;X{s)ds,
Jo
where the anticipation comes from the initial condition. The 5-transform
can be used to derive the solution
X{t) = sgn (B(l) - 1)
Note that X{t) is not continuous in t. Consider another interesting antici
pating stochastic integral equation (see Example 13.35)
X(i) = l+ fd:{B{l)X{s))ds,
Jo
where the anticipation comes from the integrand. We prove an anticipating
Ito’s formula in Theorem 13.21 which can be used to find the solution
X(t) = exp [b (1) ji* dB{s) - ^B(l)2(l - e~^*) - i ].
Preface
On the other hand, we define white noise integrals as Pettis or Bochner
integrals taking values in the space (£^)^ of generalized functions. For ex
ample, Donsker’s delta function can be represented as a white noise integral
(see Example 13.9). This representation is applied to Feynman integrals in
Chapter 14. An existence and uniqueness theorem for white noise integral
equations is given in Theorem 13.43.
The third objective is to give an easy presentation of white noise dis
tribution theory. We have provided motivations and numerous examples.
More importantly, many new ideas and techniques are introduced. For ex
ample, the Wick tensor : x®” : is usually defined by induction. However,
being motivated by Hermite polynomials with a parameter, we find it much
easier to define : directly by
/ n\
:x
k=0 ^ ^
where r is the trace operator. This definition immediately gives the rela
tionship between Wick tensors and Hermite polynomials with a parameter.
The Kubo-Yokoi theorem says that every test function has a continuous ver
sion. A beginner in white noise distribution theory usually finds the proof
of this theorem very hard to understand. We give a rather transparent
proof of this theorem by showing that a test function (p can be represented
as in Equation (6.11) by
(f{x) =
where the renormalization : is a generalized function in (5)* for each
X and 0 is a continuous linear operator from (£)/j into itself for any /?. In
fact, this representation also yields the analytic extension of p and produces
various norm estimates. We have also introduced new ideas and techniques
in the proof of Kondratiev-Streit characterization theorem for generalized
functions in (5)^.
This book is accessible to anyone with a first year graduate course in real
analysis and some knowledge of Hilbert spaces. Prior familiarity with nu
clear spaces and Gel’fand triples is not required. The reader can easily pick
up the necessary background from Chapter 2. Anyone interested in learning
the mathematical theory of white noise can quickly do so from Chapters 3
to 9. Applications of white noise in integral kernel operators, Fourier trans
forms, Laplacian operators, white noise integration, and Feynman integrals
are given in Chapters 10 to 14. Although there are some relationships
among them, these five chapters can be read independently after Chapter
9. The material in Chapter 15 can be read right after Chapter 8. We have
put it after Chapter 13 since white noise integral equations provide a natural
motivation for studying positive generalized functions. In Appendix A we
Preface
give bibliographical notes and comment on several open problems relating
to other fields. Appendix B contains a collection of miscellaneous formulas
which are needed for this book. Some of them are actually proved as lem
mas or derived in examples. They are put together for the convenience of
the reader.
Since this book is intended to be an introduction to white noise distribu
tion theory, we can not cover every aspect of its applications. For example,
we have omitted topics such as quantum probability and Yang-Mills equa
tions, infinite dimensional rotation groups, stochastic partial differential
equations, stochastic variational equations, Dirichlet forms, and quantum
field theory. We only make a few comments on these topics in Appendix
A. On the other hand, we believe that the material in Chapters 3 to 9 will
provide adequate background for applications to many problems which use
white noise. In Appendix A we mention several open problems for further
research. It is our sincere hope that the reader will find white noise dis
tribution theory interesting. If the reader can apply this theory to other
fields, it will be our greatest pleasure.
This book is based on a series of lectures which I gave at the Daewoo
Workshop and at several universities in Korea (Seoul National University,
Sogang University, Sook Myung Women’s University, Korean Advanced
Institute of Science and Technology, Chonnam National University, and
Yonsei University) during the summer of 1994. Toward the end of my visit
Professor D. M. Chung suggested that I should write up my lecture notes,
which eventually became the manuscript for this book. Were it not for
his suggestion, this book would not have materialized. I certainly was not
thinking about writing a book before my visit to Korea.
My visit to Korea was arranged by Professor D. M. Chung with finan
cial support from Daewoo Foundation, Global Analysis Research Center at
Seoul National University, Sogang University, and Sook Myung Women’s
University. I would like to express my gratitude to him and these institu
tions. I want to express my appreciation for the warm hospitality of the
following persons: K. S. Chang, S. J. Chang, D. P. Chi, S. M. Cho, B. D.
Choi, T. S. Chung, U. C. Ji, S. H. Kang, S. J. Kang, D. H. Kim, S. K. Kim,
K. Sim Lee, Y. M. Park, Y. S. Shim, I. S. Wee, and M. H. Woo.
I want to take this opportunity to thank each of the following persons and
their institutions for inviting me to give a series of lectures: B. D. Choi (Ko
rean Advanced Institute of Science and Technology), P. L. Chow (Wayne
State University), T. Hida (Meijo University and Nagoya University), C.
R. Hwang (Academia Sinica), Y.-J. Lee (Cheng Kung University), T. F.
Lin (Soochow University), Y. Okabe (Hokkaido University), R. Rebolledo
(Pontificia Universidad Católica de Chile), H. Sato (Kyushu University),
N. R. Shieh (Taiwan University), L. Streit (Universität Bielefeld), and Y.
Zhang (Fudan University). The lecture notes resulting from these visits
have infiuenced the writing of this book.
