Generalized Mehler Semigroup on White Noise Functionals and White Noise Evolution Equations

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.

Keywords

• white noise theory
• Gaussian space
• generalized Fourier–Gauss transform
• generalized Fourier–Mehler transform
• generalized Mehler semigroup
• evolution equation