Time regularity for generalized Mehler semigroups

TL;DR

This paper investigates the regularity properties of generalized Mehler semigroups, focusing on their continuity and Hölder continuity in the space of bounded continuous functions over Banach spaces, with applications to various differential operators.

Contribution

It provides a comprehensive analysis of the regularity of semigroups generated by differential and pseudo-differential operators in finite and infinite dimensions.

Findings

Established conditions for continuity and Hölder continuity of $t o P_t f$

Applied results to Ornstein-Uhlenbeck operators with fractional diffusion

Extended theory to infinite-dimensional strong-Feller semigroups

Abstract

We study continuity and H\"older continuity of $t\mapsto P_tf$, where $P_t$ is a generalized Mehler semigroup in $C_b(X)$, the space of the continuous and bounded functions from a Banach space $X$ to $R$, and $f\in C_b(X)$. The generators $L$ of such semigroups are realizations of a class of differential and pseudo-differential operators, both in finite and in infinite dimension. Examples of operators $L$ to which this theory is applicable include Ornstein-Uhlenbeck operators with fractional diffusion in finite dimension, and Ornstein-Uhlenbeck operators with associated strong-Feller semigroups, in infinite dimension.