Generalized Mehler semigroups: The non-Gaussian case
A generalized Mehler semigroup on a Hilbert space \(E\), acting on bounded Borel functions \(f:E\to{\mathbb R}\), is defined by the formula \[ p_t f(x) = \int_E f(T_tx+y)\mu_t(dy), \quad x\in E,\;t > 0. \] Here \((T_t)_{t\geq 0}\) is a strongly continuous semigroup of linear operators on \(E\) and \(\mu_t\) is a family of probability measures on \(E\). These semigroups were introduced by \textit{V. I. Bogachev, M. Röckner} and \textit{B. Schmuland} [Probab. Theory Relat. Fields 105, No. 2, 193-225 (1996; Zbl 0849.60066)] and studied especially in the Gaussian case. In the present paper the authors consider the general, non-Gaussian case. In particular, an associated Markov process is constructed, tightness of the associated \((r,p)\)-capacities is proved. Moreover, existence of invariant measures, strong Feller property, time regularity in the space of uniformly continuous and bounded functions, and description of the generator are also studied.
scientific article; zbMATH DE number 1436912