报告题目：Ergodicity andExponential Ergodicity of Feller-Markov Processes on Infinite DimensionalPolish Spaces
报 告 人：巩馥洲教授(中国科学院)
There exists a long literature of studyingthe ergodicity and asymptotic stability for various semigroups from dynamicsystems and Markov chains. Abundant theories and applications have beenestablished for compact or locally compact state spaces. However, it seems veryhard to extend all of them to infinite dimensional or general Polish settings.Actually, in the field of stochastic partial differential equations, theuniqueness of ergodic measures can be derived from the strong Feller propertybesides topological irreducibility, which has been a routine to deal with theequations with non-degenerate additive noise. The solutions of these kinds ofstochastic partial differential equations just determine the strong Feller semigroupson some Banach spaces, there exist the transition densities with respect to theergodic measures, and hence the semigroups consisting of compact operatorsunder very week integrable conditions with respect to the ergodic measures.Recent years, people have developed many new approaches to more complicatedmodels. For instance, the asymptotic strong Feller property, as a celebratingbreakthrough, was presented by Hairer and Mattingly, which can be applied todeal with the uniqueness of ergodicity for 2D Navier-Stokes equations withdegenerate stochastic forcing. Some notable contributions to this subject camefrom Lasota and Szarek along with their sequential works for equicontinuoussemigroups. Indeed, the equicontinuity is adaptable to many known stochasticpartial differential equations containing the 2D Navier-Stokes equations withdegenerate stochastic forcing. However, on one hand, it seems far from beingnecessary in the theoretical sense. On the other hand, there also existnon-equicontinuous semigroups, or it is too complicated to prove theirequicontinuity. For example, for the Ginzburg-Landau interface modelintroduced by Funaki and Spohn, it seems hopeless to prove the equicontinuity.In this talk, we will give the sharp criterions or quivalent characterizationsabout the ergodicity and asymptotic stability for Feller semigroups on Polishspaces with full generality. To this end we will introduce some new notions,especially the eventual continuity of Feller semigroups, which seems very closeto be necessary for the ergodic behavior in some sense and also allows thesensitive dependence on initial data in some extent. Furthermore, we willrevisit the unique ergodicity and prove the asymptotic stability of stochastic2D Navier-Stokes equations with degenerate stochastic forcing according to ourcriteria.