Non-Gaussian infinite dimensional analysis
type: Item
instance of: scholarly article
title: Non-Gaussian infinite dimensional analysis
zbMATH Open document ID: 0868.60041
published in: Journal of Functional Analysis
publication date: 1997-08-18T00:00:00Z
review text: White noise analysis, initiated by T. Hida, is based on a Gaussian probability space \((\Phi',\mu)\), where \(\Phi'\) is the dual of a nuclear countable Hilbertian space \(\Phi\) (e.g. the Schwartz distribution space), and \(\mu\) is a Gaussian measure, called white noise measure, with characteristic functional \(\int e^{\langle\varphi,x\rangle}\mu(dx)=\exp\{-{1\over 2}|\varphi|^2\}\). It is well-known that square-integrable Wiener functionals, i.e. the elements of \(L^2(\Phi',\mu)\), have chaotic expansions, which can be constructed by using Hermite polynomials. In terms of chaotic expansions, test functionals and so-called Hida distributions are introduced. A Hida distribution can be characterized by its \(S\)-transform. A fundamental result is the characterization theorem for \(S\)-transforms on Hida distributions, established by Potthoff and Streit. The comprehensive investigation on white noise analysis refers to the book of \textit{T. Hida}, \textit{H.-H. Kuo}, \textit{J. Potthoff} and the fourth author [``White noise: An infinite dimensional calculus'' (1993; Zbl 0771.60048)]. The paper extends the framework of white noise analysis to non-Gaussian case. The Gaussian measure is replaced by a smooth measure \(\mu\), which admits integration by parts along \(\Phi\): \[ \int\nabla_\varphi f(x)\mu(dx)= -\int f(x)\langle\varphi,\beta(x)\rangle \mu(dx) \] for any bounded \(\Phi\)-differentiable function \(f\), where \[ \nabla_\varphi f(x)= {d\over d\varepsilon} f(x+\varepsilon\varphi)\biggl|_{\varepsilon=0},\quad\varphi\in\Phi, \] and \(\beta\) is a measurable mapping from \(\Phi'\) into itself. Under two additional analytic conditions imposing upon \(\mu\), two biorthogonal systems of polynomials are defined, and they will play the similar role as Hermite polynomials in Gaussian case. By making use of expansions into the two systems, test functions and distributions are constructed. The paper ends by a similar characterization for distributions.
Mathematics Subject Classification ID: 60G20
zbMATH DE Number: 908029
zbMATH Keywords: infinite-dimensional stochastic analysis
zbMATH Keywords: white noise analysis
zbMATH Keywords: non-Gaussian stochastic analysis
author: S. Albeverio
author: Yuri L. Daletskii
author: Yuri Kondratiev
author: L. Streit
MaRDI profile type: Publication
full work available at URL: jfan.1996.0067
OpenAlex ID: W2005904265
DOI: 10.1006/JFAN.1996.0067
description: scientific article; zbMATH DE number 908029