Couplings, generalized couplings and uniqueness of invariant measures

TL;DR

This paper establishes new sufficient conditions for the uniqueness of invariant measures of Markov kernels using generalized couplings, extending previous results beyond Polish spaces and highlighting limitations in non-Polish separable metric spaces.

Contribution

It generalizes existing theorems on invariant measure uniqueness to broader spaces and demonstrates the necessity of Polish space assumptions with a counterexample.

Findings

Provided new sufficient conditions for uniqueness using generalized couplings.

Extended the theory to non-Polish separable metric spaces.

Showed that uniqueness can fail without Polish space assumptions.

Abstract

We provide sufficient conditions for uniqueness of an invariant probability measure of a Markov kernel in terms of (generalized) couplings. Our main theorem generalizes previous results which require the state space to be Polish. We provide an example showing that uniqueness can fail if the state space is separable and metric (but not Polish) even though a coupling defined via a continuous and positive definite function exists.