Asymptotic stability and mean ergodicity of Feller processes on Polish spaces

TL;DR

This paper develops necessary and sufficient criteria for $ au$-asymptotic stability and $ au$-mean ergodicity of Feller processes on Polish spaces under stronger topologies such as Wasserstein and weighted total variation. It frames the characterizations in terms of generalized eventual continuity ($\mathrm{EvC}$) and lower bound conditions ($\mathrm{LBC}$) with a test-family $\frak{F}$, and proves equivalences via a coupling-based approach, including localized regularity at a point $z$ and uniform versions for coupling distances. The results cover both standard and Cesàro ergodic notions, require hypotheses $(\mathbf{H_1})$ and $(\mathbf{H_2})$ to handle unbounded test-functions, and encompass applications to $W_p$-convergence, weighted TV, and equicontinuous processes, with explicit illustrations such as the stochastic Navier–Stokes equation. By connecting ergodic behavior in strong topologies to verifiable regularity and integrability conditions, the work provides practical criteria for long-time convergence in diverse infinite-dimensional settings.

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.

Theorems & Definitions (64)

• Definition 1.1
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Definition 2.6
• Theorem 2.7
• Remark 2.8
• Proposition 2.9