Infinite-dimensional stochastic Cauchy problems with white noise processes in spaces of distributions

2016 ◽  
pp. 197-228
Keyword(s):  
White Noise ◽  
Cauchy Problems ◽  
Infinite Dimensional

scholarly journals Transformations on white noise functions associated with second order differential operators of diagonal type

1998 ◽  
Vol 149 ◽  
pp. 173-192 ◽  
Author(s):  
Dong Myung Chung ◽  
Un Cig Ji ◽  
Nobuaki Obata
Keyword(s):  
White Noise ◽  
Differential Operators ◽  
Transformation Groups ◽  
Cauchy Problems ◽  
Noise Distribution ◽  
Number Operator ◽  
Parameter Transformation ◽  
Infinite Dimensional ◽  
White Noise Distribution Theory ◽  
White Noise Distribution

Abstract.A generalized number operator and a generalized Gross Laplacian are introduced on the basis of white noise distribution theory. The equicontinuity is examined and associated one-parameter transformation groups are constructed. An infinite dimensional analogue of ax + b group and Cauchy problems on white noise space are discussed.

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 White noise delta functions and continuous version theorem

1993 ◽  
Vol 129 ◽  
pp. 1-22
Author(s):  
Nobuaki Obata
Keyword(s):  
White Noise ◽  
Harmonic Analysis ◽  
Quantum Physics ◽  
Fock Space ◽  
Continuous Version ◽  
Infinite Dimensional ◽  
Schwartz Distribution ◽  
Delta Functions ◽  
Gelfand Triple ◽  
Gaussian Space

The recently developed Hida calculus of white noise [5] is an infinite dimensional analogue of Schwartz’ distribution theory besed on the Gelfand triple (E) ⊂ (L2) = L2 (E*, μ) ⊂ (E)*, where (E*, μ) is Gaussian space and (L2) is (a realization of) Fock space. It has been so far discussed aiming at an application to quantum physics, for instance [1], [3], and infinite dimensional harmonic analysis [7], [8], [13], [14], [15].

White noise delta functions and infinite-dimensional Laplacians

2020 ◽  
pp. 2050028
Author(s):  
Luigi Accardi ◽  
Ai Hasegawa ◽  
Un Cig Ji ◽  
Kimiaki Saitô
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Time Derivative ◽  
Delta Function ◽  
Itô Formula ◽  
Ito Formula ◽  
Infinite Dimensional ◽  
Delta Functions ◽  
Definition Of ◽  
White Noise Distribution

In this paper, we introduce a new white noise delta function based on the Kubo–Yokoi delta function and an infinite-dimensional Brownian motion. We also give a white noise differential equation induced by the delta function through the Itô formula introducing a differential operator directed by the time derivative of the infinite-dimensional Brownian motion and an extension of the definition of the Volterra Laplacian. Moreover, we give an extension of the Itô formula for the white noise distribution of the infinite-dimensional Brownian motion.

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.

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