Abstract
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.
Highlights
Since the white noise theory initiated by Hida [1] as an infinite dimensional distribution theory, it has been extensively studied by many authors [2,3,4,5,6,7,8] with many applications to wide research fields, stochastic analysis, quantum field theory, mathematical physics, mathematical finance and etc
As main results of this paper, we provide a representation of the generalized Mehler semigroup in terms of the generalized Fourier–Gauss transform on the space of the test white noise functionals, and by applying the properties of the generalized Fourier–Gauss transform, we have an explicit form of the infinitesimal generator of the generalized Mehler semigroup, which is a perturbation of the Ornstein–Uhlenbeck generator
As one of main results of this paper, in the converse direction of the Mehler’s formula, we have considered the generalized Fourier–Gauss transform as an exponential form of the generalized Mehler semigroup
Summary
Since the white noise theory initiated by Hida [1] as an infinite dimensional distribution theory, it has been extensively studied by many authors [2,3,4,5,6,7,8] (and references cited therein) with many applications to wide research fields, stochastic analysis, quantum field theory, mathematical physics, mathematical finance and etc. The objective of this paper is twofold: the first one is to study the generalized Mehler semigroup on the space ( E) of the test white noise functionals with its explicit form in terms of the generalized. As main results of this paper, we provide a representation of the generalized Mehler semigroup in terms of the generalized Fourier–Gauss transform on the space of the test white noise functionals, and by applying the properties of the generalized Fourier–Gauss transform, we have an explicit form of the infinitesimal generator of the generalized Mehler semigroup, which is a perturbation of the Ornstein–Uhlenbeck generator. We study the generalized Fourier–Mehler transform and its infinitesimal generator, which induce the dual semigroup of the generalized Mehler semigroup and its infinitesimal generator, and as application we investigate the unique weak solution of the Langevin type stochastic evolution equations with very singular noise forcing terms (see Theorem 9).
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