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 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].

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]).

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.

scholarly journals An Analytic Characterization of p , q -White Noise Functionals

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Anis Riahi ◽  
Amine Ettaieb ◽  
Wathek Chammam ◽  
Ziyad Ali Alhussain
Keyword(s):  
White Noise ◽  
Entire Functions ◽  
Fock Space ◽  
Gaussian White Noise ◽  
Characterization Theorem ◽  
Quantum Oscillator ◽  
Exponential Map ◽  
Infinite Dimensional ◽  
White Noise Functionals ◽  
Young Functions

In this paper, a characterization theorem for the S -transform of infinite dimensional distributions of noncommutative white noise corresponding to the p , q -deformed quantum oscillator algebra is investigated. We derive a unitary operator U between the noncommutative L 2 -space and the p , q -Fock space which serves to give the construction of a white noise Gel’fand triple. Next, a general characterization theorem is proven for the space of p , q -Gaussian white noise distributions in terms of new spaces of p , q -entire functions with certain growth rates determined by Young functions and a suitable p , q -exponential map.

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.

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