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Non-Gaussian infinite dimensional analysis. (English) Zbl 0868.60041

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 T. Hida, H.-H. Kuo, 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.

MSC:

60G20 Generalized stochastic processes

Citations:

Zbl 0771.60048
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