scholarly journals Lévy white noise measures on infinite-dimensional spaces: Existence and characterization of the measurable support

2006 ◽  
Vol 237 (2) ◽  
pp. 617-633 ◽  
Author(s):  
Yuh-Jia Lee ◽  
Hsin-Hung Shih
Keyword(s):  
White Noise ◽  
Infinite Dimensional ◽  
Lévy White Noise

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.

AN APPLICATION OF THE SEGAL–BARGMANN TRANSFORM TO THE CHARACTERIZATION OF LÉVY WHITE NOISE MEASURES

2010 ◽  
Vol 13 (02) ◽  
pp. 191-221 ◽  
Author(s):  
YUH-JIA LEE ◽  
HSIN-HUNG SHIH
Keyword(s):  
White Noise ◽  
Probability Measures ◽  
Infinitely Divisible Distributions ◽  
Infinite Dimensions ◽  
Infinitely Divisible ◽  
Tempered Distributions ◽  
Bargmann Transform ◽  
Second Moment ◽  
Lévy White Noise

Being inspired by the observation that the Stein's identity is closely connected to the quantum decomposition of probability measures and the Segal–Bargmann transform, we are able to characterize the Lévy white noise measures on the space [Formula: see text] of tempered distributions associated with a Lévy spectrum having finite second moment. The results not only extends the Stein and Chen's lemma for Gaussian and Poisson distributions to infinite dimensions but also to many other infinitely divisible distributions such as Gamma and Pascal distributions and corresponding Lévy white noise measures on [Formula: see text].

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.

ANALYTIC CHARACTERIZATION OF GENERALIZED FOCK SPACE OPERATORS AS TWO-VARIABLE ENTIRE FUNCTIONS WITH GROWTH CONDITION

2002 ◽  
Vol 05 (03) ◽  
pp. 395-407 ◽  
Author(s):  
UN CIG JI ◽  
NOBUAKI OBATA ◽  
HABIB OUERDIANE
Keyword(s):  
Differential Equation ◽  
White Noise ◽  
Growth Condition ◽  
Growth Rates ◽  
Entire Functions ◽  
Fock Space ◽  
Infinite Dimensional ◽  
Singular Coefficients ◽  
Young Functions

Duality is established for new spaces of entire functions in two infinite dimensional variables with certain growth rates determined by Young functions. These entire functions characterize the symbols of generalized Fock space operators. As an application, a proper space is found for a solution to a normal-ordered white noise differential equation having highly singular coefficients.

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