Relation Between the Eventual Continuity and the E-property for Markov-Feller Semigroups

We investigate some relations between two kinds of semigroup regularities, namely the e-property and the eventual continuity, both of which contribute to the ergodicity for Markov processes on Polish spaces.More precisely, we prove that for Markov-Feller semigroup in discrete time and stochastically continuous MarkovFeller semigroup in continuous time, if there exists an ergodic measure whose support has a nonempty interior,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 stochastically continuous continuous-time Markov-Feller semigroups.