# Student Probability Seminar

#### Ergodicity for Markov chains: coupling probabilities and analysis

**Speaker:**
GrÃ©goire FerrÃ©, CERMICS, ENPC

**Location:**
Warren Weaver Hall 512

**Date:**
Wednesday, December 5, 2018, noon

**Synopsis:**

Studying the long time behaviour of Markov chains is an important theoretical and practical problem. Indeed, it may be used to assess the validity of physical models such as particle systems or stochastic PDEs, as well as to control discretization schemes and their rates of convergence for Monte Carlo methods, a standard problem in statistics and Machine Learning. Such convergence results are generally refered to as "ergodicity".

There has been a number of works in this area, starting with the initial paper of Markov where ergodicity is actually proven (!) for a finite state space model. However, the general situation is much more complex, and the goal of this talk is to introduce of couple techniques for proving ergodicity. As a starter, I will show two short proofs for systems evolving in a compact state space : a probabilistic one based on coupling, and an analytic one derived by M. Hairer. I will then present a more general result with a recent proof of M. Hairer and J. Mattingly, which dramatically simplifies previous probabilistic methods. If time allows, I'll broach the situation when the evolution kernel does not preserve mass and one has to consider non-linear semigroups, a common situation in large deviation theory. From a broader perspective, my goal is to give a hint of the interesting interplay between probability and analysis in the study of Markov chains.