L2-Wasserstein contraction for Euler schemes of elliptic diffusions and interacting particle systems

We show $L^2$-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 $L^1$-Wasserstein contraction, or $L^p$-Wasserstein bounds for $p>1$ that were, however, not true contractions. We explain how showing a true $L^2$-Wasserstein contraction is crucial for obtaining a local Poincaré 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 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 optimisation literature.