Transportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and coupling

By using the mirror coupling for solutions of SDEs driven by pure jump L\'evy processes, we extend some transportation and concentration inequalities, which were previously known only in the case where the coefficients in the equation satisfy a global dissipativity condition. Furthermore, by using the mirror coupling for the jump part and the coupling by reflection for the Brownian part, we extend analogous results for jump diffusions. To this end, we improve some previous results concerning such couplings and show how to combine the jump and the Brownian case. As a crucial step in our proof, we develop a novel method of bounding Malliavin derivatives of solutions of SDEs with both jump and Gaussian noise, which involves the coupling technique and which might be of independent interest. The bounds we obtain are new even in the case of diffusions without jumps.Comment: 40 pages, revised version, accepted for publication in Annales de l'Institut Henri Poincar\'e Probabilit\'es et Statistiques. The final manuscript is available at Project Euclid via https://projecteuclid.org/euclid.aihp/157320362

Coupling and exponential ergodicity for stochastic differential equations driven by L\'{e}vy processes

We present a novel idea for a coupling of solutions of stochastic differential equations driven by L\'{e}vy 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.Comment: 40 pages, revised version, accepted for publication in Stochastic Processes and their Applications. The final manuscript is available at Elsevier via https://doi.org/10.1016/j.spa.2017.03.02

Quantitative contraction rates for Markov chains on general state spaces

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 $\mathbb{R^d}$ 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.Comment: 39 page

Exponential ergodicity for SDEs and McKean-Vlasov processes with L\'{e}vy noise

We study stochastic differential equations (SDEs) of McKean-Vlasov type with distribution dependent drifts and driven by pure jump L\'{e}vy processes. We prove a uniform in time propagation of chaos result, providing quantitative bounds on convergence rate of interacting particle systems with L\'{e}vy noise to the corresponding McKean-Vlasov SDE. By applying techniques that combine couplings, appropriately constructed $L^1$-Wasserstein distances and Lyapunov functions, we show exponential convergence of solutions of such SDEs to their stationary distributions. Our methods allow us to obtain results that are novel even for a broad class of L\'{e}vy-driven SDEs with distribution independent coefficients.Comment: 36 page

L2L^2-Wasserstein contraction for Euler schemes of elliptic diffusions and interacting particle systems

We show the $L^2$-Wasserstein contraction for the transition kernel of a discretised diffusion process, under a contractivity at infinity condition on the drift and a sufficiently high diffusivity requirement. This extends recent results that, under similar assumptions on the drift but without the diffusivity restrictions, showed the $L^1$-Wasserstein contraction, or $L^p$-Wasserstein bounds for $p > 1$ that were, however, not true contractions. We explain how showing the true $L^2$-Wasserstein contraction is crucial for obtaining the local Poincar\'{e} inequality for the transition kernel of the Euler scheme of a diffusion. Moreover, we discuss other consequences of our contraction results, such as concentration inequalities and convergence rates in KL-divergence and total variation. We also study the corresponding $L^2$-Wasserstein contraction for discretisations of interacting diffusions. As a particular application, this allows us to analyse the behaviour of particle systems that can be used to approximate a class of McKean-Vlasov SDEs that were recently studied in the mean-field optimization literature.Comment: 28 page

Nonasymptotic bounds for sampling algorithms without log-concavity

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 L2 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 log-concavity at infinity condition. We provide novel L2 convergence rates for Euler schemes, expressed explicitly in terms of problem parameters. From there we derive nonasymptotic 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 L1 and L2 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