Asymptotic stability and mean ergodicity of Feller processes on Polish spaces

TL;DR

This paper provides necessary and sufficient conditions for asymptotic stability and mean ergodicity of Feller processes on Polish spaces, using coupling methods and various topologies like Wasserstein and total variation.

Contribution

It introduces new criteria based on generalized continuity and lower bound conditions for stability and ergodicity in different topologies.

Findings

Criteria for asymptotic stability in Wasserstein distance

Criteria for mean ergodicity in weighted total variation

Use of coupling approach for proofs

Abstract

This article establishes several necessary and sufficient criteria on asymptotic stability and mean ergodicity in various types of topologies for Feller processes taking values in Polish spaces. In particular, asymptotic stability and mean ergodicity in Wasserstein distance and weighted total variation distance are considered. The characterizations are formulated by using the notions of generalized eventual continuity properties and lower bound conditions, where the proofs invoke the coupling approach.