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 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.