Relation between the eventual continuity and the e-property for Markov-Feller semigroups

TL;DR

This paper explores the relationship between the e-property and eventual continuity in Markov-Feller semigroups, establishing conditions under which these properties are equivalent on the support of ergodic measures.

Contribution

It proves that under certain conditions, the e-property and eventual continuity are equivalent for Markov-Feller semigroups with ergodic measures.

Findings

e-property holds on the interior of the ergodic measure's support

e-property and eventual continuity are equivalent on the support

results apply to both discrete-time and continuous-time semigroups

Abstract

We investigate the relation between the e-property and the eventual continuity, or called the asymptotic equicontinuity, which is a generalization of the e-property. We prove that, for any discrete-time or strongly continuous continuous-time eventually continuous Markov-Feller semigroup with an ergodic measure, if the interior of the support of the ergodic measure is nonempty, then the e-property is satisfied on the interior of the support. In particular, it implies that, restricted on the support of each ergodic measure, the e-property and the eventual continuity are equivalent for the discrete-time and the strongly continuous continuous-time Markov-Feller semigroups.