Volume 6 (2023)
Monmarché, Pierre
Wasserstein contraction and Poincaré inequalities for elliptic diffusions with high diffusivity
Annales Henri Lebesgue, Volume 6 (2023), pp. 941-973.

Metadata

DOI10.5802/ahl.182
Keywords Wasserstein distance, Poincaré inequality, coupling methods, mean-field interaction, propagation of chaos, McKean–Vlasov processes

Abstract

We consider elliptic diffusion processes on ℝd. Assuming that the drift contracts distances outside a compact set, we prove that, when the diffusion coefficient is sufficiently large, the Markov semi-group associated to the process is a contraction of the 𝒲2 Wasserstein distance, which implies a Poincaré inequality for its invariant measure. The result doesn’t require neither reversibility nor an explicit expression of the invariant measure, and the estimates have a sharp dependency on the dimension. Some variations of the arguments are then used to study, first, the stability of the invariant measure of the process with respect to its drift and, second, systems of interacting particles, yielding a criterion for dimension-free Poincaré inequalities and quantitative long-time convergence for non-linear McKean–Vlasov type processes.


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