Asymptotic strong Feller property and local weak irreducibility via generalized couplings

TL;DR

This paper demonstrates how generalized couplings can establish the asymptotic strong Feller property and local weak irreducibility, leading to the uniqueness of invariant measures in stochastic processes.

Contribution

It introduces a novel approach using generalized couplings to prove key properties related to invariant measures, extending previous results by Hairer and Mattingly.

Findings

Generalized couplings can establish ASF and local weak irreducibility.

A stronger ASF combined with local weak irreducibility guarantees uniqueness of invariant measures.

The results are shown to be optimal in a certain sense.

Abstract

In this short note we show how the asymptotic strong Feller property (ASF) and local weak irreducibility can be established via generalized couplings. We also prove that a stronger form of ASF together with local weak irreducibility implies uniqueness of an invariant measure. The latter result is optimal in a certain sense and complements some of the corresponding results of Hairer, Mattingly (2008).