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

Linshan Liu; Mateusz B. Majka; Pierre Monmarché

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arXiv · 2310.15897

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

Linshan Liu·Mateusz B. Majka·Pierre Monmarché

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Abstract

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 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 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.

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BibTeX RIS

Linshan Liu, Mateusz B. Majka, Pierre Monmarché. 2023-10-24. $L^2$-Wasserstein contraction for Euler schemes of elliptic diffusions and interacting particle systems. https://arxiv.org/abs/2310.15897

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