There exists a long literature of studying the ergodicity and asymptotic stability for various Feller-Markov semigroups from dynamic systems and Markov processes. Abundant theories and applications have been established for the compact semigroups and the semigroups on compact or locally compact state spaces. However, it seems very hard to extend all of them to infnite dimensional or general Polish settings.  In this talk, we will give the sharp criterions or equivalent characterizations about the ergodicity and asymptotic stability for Feller-Markov semigroups on Polish spaces with full generality. To this end we will introduce some new notions, especially the eventual continuity of Feller semigroups, which seems very close to be necessary for the ergodic behavior in some sense and also allows the sensitive dependence on initial data in some extent. Furthermore, we will revisit the unique ergodicity and prove the asymptotic stability of stochastic 2D Navier-Stokes equations with degenerate stochastic forcing according to our criteria.

If the Feller-Markov semigroups are asymptotic stable, how to estimate the convergence rate of it to ergodic measure? More importantly, how to estimate to the exponential convergence rate for the exponential ergodic Feller-Markov semigroups?

In general cases, we can use the so called Ricci Curvature of Markov Chains to give the estimates.

In this talk, we will introduce our some results in this topic.

If the state space of Feller-Markov semigroups are linear spaces, then the above problems are concerned with the below problem: how to use the information of coeffcients in the corresponding partial differential operators as the generator to get the information of spectrum of the operators? In particular, spectral gap of the operators concern with the exponential ergodicity of the corresponding Feller-Markov semigroups.

In this talk we will extend the fundamental gap comparison theorem of Andrews and Clutterbuck to the infnite dimensional setting. Furthermore, we will give the probabilistic proofs of fundamental gap conjecture and spectral gap comparison theorem of Andrews and Clutterbuck in fnite dimensional case via the coupling by refection of the difusion processes.

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