On the existence of an invariant measure for Markov?Feller operators

Abstract

Let X be a Polish space and P a Markov operator acting on the space of Borel measures on X . We will prove the existence of an invariant measure with respect to P , provided that P satisfies some condition of a Prokhorov type and that the family of functions {x↦P n δ x : n∈ N } is equi-continuous with respect to the Prokhorov distance at some point of the space X . Moreover, we will construct a counterexample which show that the above equi-continuity condition cannot be dropped.