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 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 Ito’s formula and Levy’s Laplacian II

1991 ◽  
Vol 123 ◽  
pp. 153-169 ◽  
Author(s):  
Kimiaki Saito
Keyword(s):  
White Noise ◽  
Essential Role ◽  
Beltrami Operator ◽  
Itô’S Formula ◽  
Infinite Dimensional ◽  
Princeton University ◽  
Itô's Formula ◽  
Ito’S Formula ◽  
White Noise Functionals ◽  
White Noise Calculus

The white noise calculus was initiated by T. Hida in 1970 in his Princeton University Mathematical Notes [3]. Recent development of the theory shows that the Laplacian plays an essential role in the analysis in question. Indeed, several kinds of Laplacians should be introduced depending on the choice of the class of white noise functionals to be analysed, as can be seen in [4], [13], [18] and so forth. Among others, we should like to emphasize the importance of the infinite dimensional Laplace-Beltrami operator, Volterra’s Laplacian and Lévy’s Laplacian (See [13], [18] and [20]).

scholarly journals Derivations on white noise functionals

1995 ◽  
Vol 139 ◽  
pp. 21-36 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
White Noise ◽  
Topological Algebra ◽  
Fundamental Question ◽  
Special Choice ◽  
Pointwise Multiplication ◽  
Infinite Dimensional ◽  
White Noise Functionals ◽  
White Noise Calculus ◽  
Gelfand Triple ◽  
Gaussian Space

The Gaussian space (E*, μ) is a natural infinite dimensional analogue of Euclidean space with Lebesgue measure and a special choice of a Gelfand triple gives a fundamental framework of white noise calculus [2] as distribution theory on Gaussian space. It is proved in Kubo-Takenaka [7] that (E) is a topological algebra under pointwise multiplication. The main purpose of this paper is to answer the fundamental question: what are the derivations on the algebra (E)?

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.

Analytic characterizations of infinite dimensional distributions

2017 ◽  
Vol 20 (02) ◽  
pp. 1750007 ◽  
Author(s):  
Luigi Accardi ◽  
Un Cig Ji ◽  
Kimiaki Saitô
Keyword(s):  
White Noise ◽  
Characterization Theorem ◽  
Growth Conditions ◽  
Necessary Condition ◽  
Infinite Dimensional ◽  
S Transform ◽  
White Noise Functionals

We revisit the analytic characterization theorem for S-transform of infinite dimensional distributions. Then we prove that the nuclearity of the space of test white noise functionals is a necessary condition for the characterization of the S-transform in terms of analytic and growth conditions.

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 Generalized Mehler Semigroup on White Noise Functionals and White Noise Evolution Equations

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1025
Author(s):  
Un Cig Ji ◽  
Mi Ra Lee ◽  
Peng Cheng Ma
Keyword(s):  
Evolution Equation ◽  
White Noise ◽  
Explicit Form ◽  
Infinitesimal Generator ◽  
Evolution Equations ◽  
Time Derivative ◽  
White Noise Functionals

In this paper, we study a representation of generalized Mehler semigroup in terms of Fourier–Gauss transforms on white noise functionals and then we have an explicit form of the infinitesimal generator of the generalized Mehler semigroup in terms of the conservation operator and the generalized Gross Laplacian. Then we investigate a characterization of the unitarity of the generalized Mehler semigroup. As an application, we study an evolution equation for white noise distributions with n-th time-derivative of white noise as an additive singular noise.

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