Georgia Institute of Technology College of Sciences

In my talk, I will present the main results of a recent article by Martin Hairer and Jonathan Mattingly on an ergodic theorem for Markov chains. I will consider Markov chains evolving in discrete time on an abstract, possibly uncountable, state space. Under certain regularity assumptions on the chain's transition kernel, such as the existence of a Foster-Lyapunov function with small level sets (what exactly is meant by that will be thoroughly explained in the talk), one can establish the existence and uniqueness of a stationary distribution. I will focus on a new proof technique for that theorem which relies on a family of metrics on the set of probability measures living on the state space. The main result of my talk will be a strict contraction estimate involving these metrics.

The talk is based on the paper by B. Klartag. It will be shown that there exists a sequence \eps_n\to 0 for which the following holds: let K be a compact convex subset in R^n with nonempty interior and X a random vector uniformly distributed in K. Then there exists a unit vector v, a real number \theta and \sigma^2>0 such that d_TV(, Z)\leq \eps_n where Z has Normal(\theta,\sigma^2) distribution and d_TV - the total variation distance. Under some additional assumptions on X, the statement is true for most vectors v \in R^n.

The talk is based on the recent paper by M.Hairer, J.Mattingly, and M.Scheutzow with the same title.There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris' theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such "asymptotic couplings" were central to recent work on SPDEs on which this work builds. The emphasis here is on stochastic differential delay equations.Harris' celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are "small" (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a "small set" by the much weaker notion of a "d-small set," which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris' theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations.

The talk is based on the paper titled "Anosov diffeomorphisms and coupling" by Bressaud and Liverani. Existence and uniqueness of SRB invariant measure for the dynamics is established via a coupling of initial conditions introduced to dynamics by L.-S. Young.