Asymptotic behavior of strong Feller semigroups

TL;DR

This paper proves that weakly ergodic, strong Feller semigroups on measures converge strongly to a projection onto their fixed space without requiring stochastic continuity, extending previous results.

Contribution

It establishes convergence of such semigroups without the assumption of stochastic continuity, broadening the understanding of their asymptotic behavior.

Findings

Semigroups converge strongly to a fixed space projection

No need for stochastic continuity assumption

Extends previous ergodic convergence results

Abstract

We prove that a weakly ergodic, strong Feller semigroup on the space of measures converges strongly to a projection onto its fixed space. In contrast to a recent result of Gerlach we do not assume the semigroup to be stochastically continuous.