报告题目：Ergodicity and Exponential Ergodicity of Feller-Markov Processes on Infinite Dimensional Polish Spaces

报 告 人：巩馥洲教授(中国科学院)

报告时间：2017年5月20日下午3:30-4:30

报告地点：知新楼B1238室

报告摘要： There exists a long literature of studying the ergodicity and asymptotic stability for various semigroups from dynamic systems and Markov chains. Abundant theories and applications have been established for compact or locally compact state spaces. However, it seems very hard to extend all of them to infinite dimensional or general Polish settings. Actually, in the field of stochastic partial differential equations, the uniqueness of ergodic measures can be derived from the strong Feller property besides topological irreducibility, which has been a routine to deal with the equations with non-degenerate additive noise. The solutions of these kinds of stochastic partial differential equations just determine the strong Feller semigroups on some Banach spaces, there exist the transition densities with respect to the ergodic measures, and hence the semigroups consisting of compact operators under very week integrable conditions with respect to the ergodic measures. Recent years, people have developed many new approaches to more complicated models. For instance, the asymptotic strong Feller property, as a celebrating breakthrough, was presented by Hairer and Mattingly, which can be applied to deal with the uniqueness of ergodicity for 2D Navier-Stokes equations with degenerate stochastic forcing. Some notable contributions to this subject came from Lasota and Szarek along with their sequential works for equicontinuous semigroups. Indeed, the equicontinuity is adaptable to many known stochastic partial differential equations containing the 2D Navier-Stokes equations with degenerate stochastic forcing. However, on one hand, it seems far from being necessary in the theoretical sense. On the other hand, there also exist non-equicontinuous semigroups, or it is too complicated to prove their equicontinuity. For example, for the Ginzburg-Landau interface model introduced by Funaki and Spohn, it seems hopeless to prove the equicontinuity. In this talk, we will give the sharp criterions or quivalent characterizations about the ergodicity and asymptotic stability for Feller 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.

欢迎各位老师同学积极参加！