The asymptotic strong Feller property does not imply the e-property for Markov-Feller semigroups

TL;DR

This paper demonstrates that the asymptotic strong Feller property does not necessarily imply the e-property for Markov-Feller semigroups, providing simple counterexamples to clarify their relationship.

Contribution

It provides the first explicit counterexamples showing that the asymptotic strong Feller property does not imply the e-property and vice versa in Markov-Feller semigroups.

Findings

Counterexample showing asymptotic strong Feller does not imply e-property.

Counterexample showing e-property does not imply asymptotic strong Feller.

Clarification of the relationship between these properties in Markov semigroups.

Abstract

T. Szarek [Stud. Math. 189 (2008), {\S} 4] have discussed the relationship between two important notions concerning Markov semigroups: the asymptotic strong Feller property and the e-property, asserting that the former property implies the latter one. In this very short note we rectify this issue exhibiting a simple example of a Markov-Feller semigroup enjoying the asymptotic strong Feller property, for which the e-property is not satisfied. (See also the comment on a connection between the asymptotic strong Feller property and the e-property by T. Szarek, D. Worm [ETDS 32 (2012), {\S} 1]). Additionally we give a very simple example - in comparing with the one given by T. Szarek [Stud. Math. 189 (2008), {\S} 4] - showing that also the converse implication does not hold.