scholarly journals A characterization of the Lévy Laplacian in terms of infinite dimensional rotation groups

1990 ◽  
Vol 118 ◽  
pp. 111-132 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
Variational Calculus ◽  
Singular Part ◽  
White Noise Analysis ◽  
Infinite Dimensional ◽  
Functional Derivatives ◽  
Lévy Laplacian ◽  
Rotation Groups

P. Lévy introduced, in his celebrated books [21] and [22], an infinite dimensional Laplacian called the Lévy Laplacian in connection with a number of interesting topics in variational calculus. One of the most significant features of the Lévy Laplacian is observed when it acts on the singular part of the second functional derivatives. For this reason the Lévy Laplacian has become important also in white noise analysis initiated by T. Hida [12]. On the other hand, as was pointed out by H. Yoshizawa [29], infinite dimensional rotation groups are profoundly concerned with the structure of white noise, and therefore, play essential roles in certain problems of stochastic calculus. Motivated by these works, we aim at developing harmonic analysis on infinite dimensional spaces by means of the Lévy Laplacian and infinite dimensional rotation groups.

scholarly journals Infinite dimensional rotations and Laplacians in terms of white noise calculus

1992 ◽  
Vol 128 ◽  
pp. 65-93 ◽  
Author(s):  
Takeyuki Hida ◽  
Nobuaki Obata ◽  
Kimiaki Saitô
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Quantum Physics ◽  
Variational Calculus ◽  
Rotation Group ◽  
Infinite Dimensional ◽  
Lévy Laplacian ◽  
White Noise Functionals ◽  
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.

scholarly journals A note on certain permutation groups in the infinite dimensional rotation group

1988 ◽  
Vol 109 ◽  
pp. 91-107 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Functional Analysis ◽  
White Noise ◽  
Noise Analysis ◽  
Rotation Group ◽  
Permutation Groups ◽  
White Noise Analysis ◽  
Natural Numbers ◽  
Infinite Dimensional ◽  
Lévy Laplacian

In his book P. Lévy discussed certain permutation groups of natural numbers in connection with the theory of functional analysis. Among them the group , called the Lévy group after T. Hida, has been studied along with Hida’s theory of white noise analysis and has become very important keeping profound contact with the Lévy Laplacian which is an infinite dimensional analogue of the ordinary Laplacian.

Innovations for Random Fields

1998 ◽  
Vol 01 (04) ◽  
pp. 499-509 ◽  
Author(s):  
Takeyuki Hida ◽  
Si Si
Keyword(s):  
Stochastic Process ◽  
Random Field ◽  
White Noise ◽  
Random Fields ◽  
Noise Analysis ◽  
Variational Calculus ◽  
Modern Theory ◽  
White Noise Analysis ◽  
Future Directions ◽  
Probabilistic Structure

There is a famous formula called Lévy's stochastic infinitesimal equation for a stochastic process X(t) expressed in the form [Formula: see text] We propose a generalization of this equation for a random field X(C) indexed by a contour C. Assume that the X(C) is homogeneous in a white noise x, say of degree n, we can then appeal to the classical theory of variational calculus and to the modern theory of white noise analysis in order to discuss the innovation for the X(C) and hence its probabilistic structure. Some of future directions are also mentioned.

DIRICHLET FORMS IN TERMS OF WHITE NOISE ANALYSIS II: CLOSABILITY AND DIFFUSION PROCESSES

1989 ◽  
Vol 01 (02n03) ◽  
pp. 313-323 ◽  
Author(s):  
S. ALBEVERIO ◽  
T. HIDA ◽  
J. POTTHOFF ◽  
M. RÖCKNER ◽  
L. STREIT
Keyword(s):  
White Noise ◽  
Diffusion Processes ◽  
Noise Analysis ◽  
Dirichlet Forms ◽  
White Noise Analysis ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
Dimensional Diffusion ◽  
White Noise Functionals ◽  
And Diffusion

It is shown that infinite dimensional Dirichlet forms as previously constructed in terms of (generalized) white noise functionals fit into the general framework of classical Dirichlet forms on topological vector spaces. This entails that all results obtained there are applicable. Admissible functionals give rise to infinite dimensional diffusion processes.

CAUCHY PROBLEMS ASSOCIATED WITH THE LÉVY LAPLACIAN IN WHITE NOISE ANALYSIS

1999 ◽  
Vol 02 (01) ◽  
pp. 131-153 ◽  
Author(s):  
DONG MYUNG CHUNG ◽  
UN CIG JI ◽  
KIMIAKI SAITÔ
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
Type Equation ◽  
Cauchy Problems ◽  
White Noise Analysis ◽  
Uniqueness Of Solutions ◽  
Wave Type ◽  
Lévy Laplacian ◽  
White Noise Functionals

In this paper we shall discuss the existence and the uniqueness of solutions of the heat type equation and the wave type equation associated with the Lévy Laplacian acting on a domain in the space of generalized white noise functionals.

White-noise analysis in retinal physiology

1986 ◽  
Vol 4 ◽  
pp. S141-S152
Author(s):  
Masanori Sakuranaga ◽  
Yu-Ichiro Ando ◽  
Ken-Ichi Naka
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
White Noise Analysis
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