Ergodicity for eventually continuous Markov--Feller semigroups on Polish spaces

TL;DR

This paper studies the ergodic behavior of Markov--Feller semigroups on Polish spaces under weak regularity conditions, providing criteria for invariant measures and applying results to complex dynamical systems.

Contribution

It introduces new criteria linking Cesàro eventual continuity to ergodicity and invariant measures, expanding understanding of ergodic properties under minimal regularity assumptions.

Findings

Cesàro averages converge weakly to ergodic measures

Criteria for existence and uniqueness of invariant measures

Applications to complex systems like Lorenz and turbulence models

Abstract

This paper investigates the ergodicity of Markov--Feller semigroups on Polish spaces, focusing on very weak regularity conditions, particularly the Ces\`aro eventual continuity. First, it is showed that the Ces\`aro average of such semigroups weakly converges to an ergodic measure when starting from its support. This leads to a characterization of the relationship between Ces\`aro eventual continuity, Ces\`aro e-property, and weak-* mean ergodicity. Next, serval criteria are provided for the existence and uniqueness of invariant measures via Ces\`aro eventual continuity and lower bound conditions, establishing an equivalence relation between weak-* mean ergodicity and a lower bound condition. Additionally, some refined properties of ergodic decomposition are derived. Finally, the results are applied to several non-trivial examples, including iterated function systems, Hopf's turbulence model with random forces, and Lorenz system with noisy perturbations, either with or without Ces\`aro eventual continuity.