On some results on the stability of Markov operators

TL;DR

This paper establishes criteria for the existence of invariant measures and proves asymptotic stability of certain Feller semigroups on metric spaces, extending previous results in the field.

Contribution

It introduces a new criterion for invariant measures and demonstrates stability of semigroups under specific conditions, broadening the understanding of Markov operator stability.

Findings

Established a criterion for invariant measure existence.

Proved asymptotic stability of Feller semigroups.

Extended previous stability results to broader metric spaces.

Abstract

We formulate a criterion for the existence of an invariant measure for a Feller semigroup defined on a metric space with the e-property for bounded continuous functions and use it to prove the asymptotic stability of a semigroup satisfying a lower bound condition. Our results complement those of A. Lasota and J. A. Yorke in proper metric spaces and of T. Szarek in Polish spaces.