## Alain Durmus (Telecom ParisTech): New bounds for the sub-geometric convergence of Markov Chains in Wasserstein metric and application to the pre-conditionned Crank Nicholson algorithm

In [1], the authors generalize the conditions of the well-known Harris theorem for the convergence of Markov Chains in the Total Variation norm (TV) to obtain geometric ergodicity in Wasserstein distance. [2] takes inspiration of [1] and [3] to establish sub-geometric ergodicity in Wasserstein metric, but does not succeed to get rates as good as in [2] for the TV. We will present improvement of these rates which are between the ones in [2] and [3]. This result is establishes by a more probabilistic reasoning than in [1] and [2], who use more analytic techniques. We apply it to the non-linear autoregressive model in $\mathbb{R}^d$ and the function space MCMC algorithm : the pre-conditionned Crank-Nicolson algorithm. In particular, for the latter, we show that an simple Hölder condition on the log-density implies the sub-geometric ergodicity of the Markov chain produced by the algorithm in a Wasserstein distance, generalizing some conditions in [4] to have geometric one.

Joint work with G. Fort and E. Moulines

References :
[1]  M. Hairer, J.C. Mattingly, and M. Scheutzow. Asymptotic coupling and a
general form of Harris’ theorem with applications to stochastic delay equa-
tions. Probability Theory and Related Fields, 149(1-2):223–259, 2011.
[2] Oleg Butkovsky. Subgeometric rates of convergence of Markov processes in
the wasserstein metric. Annals of Applied Probability, 2012.
[3] Randal Douc, Gersende Fort, Eric  ́Moulines, and Philippe Soulier. Practical
drift conditions for subgeometric rates of convergence. Ann. Appl. Probab.,
14(3):1353–1377, 2004.
[4] M. Hairer, A.M. Stuart, and S.J. Vollmer. Spectral gaps for Metropolis-
Hastings algorithms in infinite dimensions.

Keywords : sub-geometric ergodicity, Markov Chain, pre-conditionned Crank-Nicolson algorithm.