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

Linshan Liu, Mateusz B. Majka and Pierre Monmarché

Stochastic Processes and their Applications, 2025, vol. 179, issue C

Abstract: We show L2-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 L1-Wasserstein contraction, or Lp-Wasserstein bounds for p>1 that were, however, not true contractions. We explain how showing a true L2-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 L2-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.

Date: 2025
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DOI: 10.1016/j.spa.2024.104504

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