Transition Semigroups of Banach Space Valued Ornstein-Uhlenbeck Processes

TL;DR

This paper studies the properties of transition semigroups associated with Banach space valued Ornstein-Uhlenbeck processes, linking their characteristics to the underlying semigroup and noise structure.

Contribution

It characterizes key properties of these semigroups, such as strong Feller, spectral gap, and analyticity, in relation to the behavior of the semigroup and noise space.

Findings

Characterization of strong Feller property in terms of the semigroup.

Conditions for spectral gap property based on the semigroup.

Analysis of analyticity and its relation to the noise space and restricted semigroup.

Abstract

We investigate the transition semigroup of the solution to a stochastic evolution equation $dX(t) = AX(t)dt +dW_H(t)$, $t\ge 0,$ where $A$ is the generator of a $C_0$-semigroup $S$ on a separable real Banach space $E$ and $W_H$ is cylindrical white noise with values in a real Hilbert space $H$ which is continuously embedded in $E$. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of $S$ in $H$. In particular we investigate the interplay between analyticity of the transition semigroup, $S$-invariance of $H$, and analyticity of the restricted semigroup $S_H$.