[Distinguished Lecture] Ergodicity and Exponential Ergodicity of Feller-Ｍarkov Semigroups on Infinite Dimensional Polish Spaces

地   点：  腾讯会议 APP()

报告人：  巩馥洲 研究员

单   位：  中科院数学与系统科学研究院

邀请人：  张登

备   注：  腾讯会议：983-766-262 (密码：123456) 报告人简介：巩馥洲研究员，主要从事随机分析领域的研究，国家杰出青年基金获得者，首批中组部“万人计划——百千万工程领军人才”人选，享受国务院政府特殊津贴，获第九届“陈省身数学奖”，"钟家庆"奖，香港求是科技基金会“杰出青年学者奖”。现任中国数学会副理事长兼秘书长，中国科学院数学与系统科学研究院副院长，《应用数学学报》主编，以及《Acta Mathematica Sinica》,《Communications in Mathematics and Statistics》，《Journal of Mathematical Research with Application》等多家期刊编委。

报告摘要：  

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 infinite 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 coefficients 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 infinite dimensional setting. Furthermore, we will give the probabilistic proofs of fundamental gap conjecture and spectral gap comparison theorem of Andrews and Clutterbuck in finite dimensional case via the coupling by reflection of the diffusion processes.