Abstract. We present a novel idea for a coupling of solutions of stochastic differential equations driven by Lévy noise, inspired by some results from the optimal transportation theory. Then we use this coupling to obtain exponential contractivity of the semigroups associated with these solutions with respect to an appropriately chosen Kantorovich distance. As a corollary, we obtain exponential convergence rates in the total variation and standard L 1 -Wasserstein distances.

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We investigate the problem of quantifying contraction coefficients of Markov transition kernels in Kantorovich (L 1 Wasserstein) distances. For diffusion processes, relatively precise quantitative bounds on contraction rates have recently been derived by combining appropriate couplings with carefully designed Kantorovich distances. In this paper, we partially carry over this approach from diffusions to Markov chains. We derive quantitative lower bounds on contraction rates for Markov chains on general state spaces that are powerful if the dynamics is dominated by small local moves. For Markov chains on R d with isotropic transition kernels, the general bounds can be used efficiently together with a coupling that combines maximal and reflection coupling. The results are applied to Euler discretizations of stochastic differential equations with non-globally contractive drifts, and to the Metropolis adjusted Langevin algorithm for sampling from a class of probability measures on high dimensional state spaces that are not globally log-concave.2010 Mathematics Subject Classification. 60J05, 60J22, 65C05, 65C30, 65C40.

Discrete time analogues of ergodic stochastic differential equations (SDEs) are one of the most popular and flexible tools for sampling high-dimensional probability measures. Non-asymptotic analysis in the L 2 Wasserstein distance of sampling algorithms based on Euler discretisations of SDEs has been recently developed by several authors for log-concave probability distributions. In this work we replace the log-concavity assumption with a logconcavity at infinity condition. We provide novel L 2 convergence rates for Euler schemes, expressed explicitly in terms of problem parameters. From there we derive non-asymptotic bounds on the distance between the laws induced by Euler schemes and the invariant laws of SDEs, both for schemes with standard and with randomised (inaccurate) drifts. We also obtain bounds for the hierarchy of discretisation, which enables us to deploy a multi-level Monte Carlo estimator. Our proof relies on a novel construction of a coupling for the Markov chains that can be used to control both the L 1 and L 2 Wasserstein distances simultaneously. Finally, we provide a weak convergence analysis that covers both the standard and the randomised (inaccurate) drift case. In particular, we reveal that the variance of the randomised drift does not influence the rate of weak convergence of the Euler scheme to the SDE. | • | =• , • is the Euclidean distance on R d . We will also work with a special class of L 1 -Wasserstein (pseudo) distances denoted by W f in which the Euclidean distance |x−y| is replaced by f (|x − y|) for some increasing function f : [0, ∞) → [0, ∞). Namely, we put W f (µ, ν) := inf π∈Π(µ,ν) R d ×R d f (|x − y|) π(dx dy). We remark that Wasserstein distances are typically preferred metrics when quantifying the quality of sampling methods, see [ACB17, DK17, GDVM16].Convergence in Wasserstein distances is typically investigated under the contractivity condition on the drift, i.e., under the assumption that there exists a constant K > 0 such thatthis condition corresponds to strong convexity of U , whereas a probability measure µ such that µ(dx) ∝ exp(−U (x))dx is then called logconcave. Convergence analysis for several sampling algorithms under condition (1.3) and the Lipschitz continuity of the drift has been recently performed in the L 2 -Wasserstein distance in papers such as [DK17, BFFN17, DM16]. In this work instead of (1.3) we work with the following assumptions. Assumptions 1.1 (Contractivity at infinity). Function b : R d → R d satisfies the following conditions: i) Lipschitz condition: there is a constant L > 0 such that |b(x) − b(y)| ≤ L|x − y| for all x, y ∈ R d . (1.4) ii) Contractivity at infinity condition: there exist constants K, R > 0 such that x − y, b(x) − b(y) ≤ −K|x − y| 2 for all x, y ∈ R d with |x − y| > R .

We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology [8] to calculate expectations with respect to the invariant measure of an ergodic SDE. In that context, we study the (over-damped) Langevin equations with a strongly concave potential. We show that, when appropriate contracting couplings for the numerical integrators are available, one can obtain a uniform in time estimate of the MLMC variance in contrast to the majority of the results in the MLMC literature. As a consequence, a root mean square error of O(ε) is achieved with O(ε −2 ) complexity on par with Markov Chain Monte Carlo (MCMC) methods, which however can be computationally intensive when applied to large data sets. Finally, we present a multi-level version of the recently introduced Stochastic Gradient Langevin Dynamics (SGLD) method [37] built for large datasets applications. We show that this is the first stochastic gradient MCMC method with complexity O(ε −2 |log ε| 3 ), in contrast to the complexity O(ε −3 ) of currently available methods. Numerical experiments confirm our theoretical findings.