Existence and uniqueness of invariant measures for non-Feller Markov semigroups

TL;DR

This paper establishes conditions for the existence and uniqueness of invariant measures for a broad class of Markov processes, including those with discontinuous dynamics, using measure-theoretic techniques.

Contribution

It introduces a novel measure-theoretic approach to prove uniqueness of invariant measures without relying on Harris recurrence or Lyapunov functions.

Findings

Existence of invariant measures under weak regularity assumptions.

Uniqueness guaranteed by a domination property of the resolvent kernel.

Applicable to jump processes and hybrid models with discontinuities.

Abstract

We study existence and uniqueness of invariant probability measures for continuous-time Markov processes on general state spaces. Existence is obtained from tightness of time averages under a weak regularity assumption inspired by quasi-Feller semigroups, allowing for discontinuous and non-Feller dynamics. Our main contribution concerns uniqueness. Under a natural $\psi$-irreducibility assumption, we show that the normalized resolvent kernel satisfies a domination property with respect to a reference measure. As a consequence, every invariant probability measure charges this reference measure. Since distinct ergodic invariant measures are mutually singular on standard Borel spaces, this domination property implies uniqueness whenever an invariant probability measure exists. The argument is purely measure-theoretic and does not rely on Harris recurrence, return-time estimates, or Foster--Lyapunov conditions, and applies in particular to jump processes and hybrid models with discontinuous dynamics.