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

Abstract

This paper investigates the ergodicity of Markov--Feller semigroups on Polish spaces, focusing on very weak regularity conditions, particularly the Cesà ro eventual continuity. First, it is showed that the Cesà ro average of such semigroups weakly converges to an ergodic measure when starting from its support. This leads to a characterization of the relationship between Cesà ro eventual continuity, Cesà ro e-property, and weak-* mean ergodicity. Next, serval criteria are provided for the existence and uniqueness of invariant measures via Cesà ro 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 Cesà ro eventual continuity.