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

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.