Generalized couplings and convergence of transition probabilities

TL;DR

This paper establishes conditions for the uniqueness of invariant measures and convergence of transition probabilities in Markov processes, using generalized couplings, and applies these results to certain SPDEs.

Contribution

It introduces new sufficient conditions based on generalized couplings for invariant measure uniqueness and weak convergence, extending previous ergodicity results.

Findings

Conditions ensure weak convergence of transition probabilities

Application to specific SPDEs confirms theoretical results

Strengthens understanding of ergodic behavior in Markov processes

Abstract

We provide sufficient conditions for the uniqueness of an invariant measure of a Markov process as well as for the weak convergence of transition probabilities to the invariant measure. Our conditions are formulated in terms of generalized couplings. We apply our results to several SPDEs for which unique ergodicity has been proven in a recent paper by Glatt-Holtz, Mattingly, and Richards and show that under essentially the same assumptions the weak convergence of transition probabilities actually holds true.