Takeyuki Hida

Infinite-dimensional rotations and Laplacians in terms of white noise calculus

The theory of generalized white noise functionals (white noise calculus) initiated in [2] has been considerably developed in recent years, in particular, toward applications to quantum physics, see e.g. [5], [7] and references cited therein. On the other hand, since H. Yoshizawa [4], [23] discussed an infinite dimensional rotation group to broaden the scope of an investigation of Brownian motion, there have been some attempts to introduce an idea of group theory into the white noise calculus. For example, conformal invariance of Brownian motion with multidimensional parameter space [6], variational calculus of white noise functionals [14], characterization of the Levy Laplacian [17] and so on.

On quantum theory in terms of white noise

It has often been pointed out that a much more manageable structure is obtained from quantum theory if the time parameter t is chosen imaginary instead of real. Under a replacement of t by i·t the Schrodinger equation turns into a generalized heat equation, time ordered correlation functions transform into the moments of a probability measure, etc. More recently this observation has become extremely important for the construction of quantum dynamical models, where criteria were developed by E. Nelson, by K. Osterwalder and R. Schrader and others [8] which would permit the reverse transition to real time after one has constructed an imaginary time (“Euclidean”) model.

Dirichlet forms and white noise analysis

We use the white noise calculus as a framework for the introduction of Dirichlet forms in infinite dimensions. In particular energy forms associated with positive generalized white noise functionals are considered and we prove criteria for their closability. If the forms are closable, we show that their closures are Markovian (in the sense of Fukushima).

Lectures on white noise functionals

White noise analysis is an advanced stochastic calculus that has developed extensively since three decades ago. It has two main characteristics. One is the notion of generalized white noise functionals, the introduction of which is oriented by the line of advanced analysis, and they have made much contribution to the fields in science enormously. The other characteristic is that the white noise analysis has an aspect of infinite dimensional harmonic analysis arising from the infinite dimensional rotation group. With the help of this rotation group, the white noise analysis has explored new areas of mathematics and has extended the fields of applications.

Intersection Local Times as Generalized White Noise Functionals

For any dimension we present the expansions of Brownian motion self-intersection local times in terms of multiple Wiener integrals. Suitably subtracted, they exist in the sense of generalized white noise functionals; their kernel functions are given in closed (and remarkably simple) form.

WHITE NOISE ANALYSIS AND APPLICATIONS

This article presents some of the recent development in white noise analysis as an infinite dimensional calculus.

The vacuum of the Høegh-Krohn model as a generalized white noise functional

Abstract We show that the expectations of a number of models of quantum fields are represented by generalized functionals of white noise. The energy forms associated with these functionals are studied.

White Noise Analysis and the Levy Laplacian

In line with the harmonic analysis on the space (L2)- of generalized Brownian functionals we are given the Levy’s Laplacian ΔL and discuss its roles in the causal calculus on (L2)-. There we can find interesting relations to the Levy group as well as to the Fourier transform introduced by H.-H. Kuo.

Quadratic functionals of Brownian motion

Functionals of Brownian motion can be dealt with by realizing them as functionals of white noise. Specifically, for quadratic functionals of Brownian motion, such a realization is a powerful tool to investigate them. There is a one-to-one correspondence between a quadratic functional of white noise and a symmetric L2(R2)-function which is considered as an integral kernel. By using well-known results on the integral operator we can study probabilistic properties of quadratic or certain exponential functionals of white noise. Two examples will illustrate their significance.

THE ITÔ TABLE OF THE SQUARE OF WHITE NOISE

In a recent paper 3] Accardi, Lu and Volovich have introduced the-algebra of the renormalized square of white noise (SWN). This algebra is generated by the operators B f ; B y f ; N f , whose formal expressions, in terms of the square of the standard quantum white noise b + t , b t are: B f = Z b 2 t f(t) dt; B y f = Z b y2 t f(t) dt; N f = Z b y t b t f(t) dt with the following commutation relations: where c is a strictly positive number, (f; g) = R f(t)g(t) dt (coming from renormalization), and is the vacuum vector. The SWN algebra was extended to the free case in 8]. The exponential vectors for the SWN were introduced in 5] where the connection with Boukas{Feinsilvers nite diier-ence algebra was rst noticed. As shown in 1], the deep roots of these connections are to be found in the theory of representation of current algebras and of quantum independent increment processes. This general approach has allowed to construct innnitely many representations of the SWN, inequivalent to the Fock one and to obtain similar representations for the higher powers of the white noise. In particular the results of 1] allow to reduce 1