Wasserstein contraction and Poincar\'e inequalities for elliptic diffusions at high temperature

TL;DR

This paper proves that elliptic diffusion processes at high temperature exhibit Wasserstein contraction and satisfy Poincaré inequalities without requiring reversibility, with results applicable to high-dimensional and interacting particle systems.

Contribution

It establishes Wasserstein contraction and Poincaré inequalities for elliptic diffusions at high temperature without reversibility or explicit invariant measures, with dimension-dependent estimates.

Findings

Wasserstein contraction holds at high temperature for certain diffusions.

Poincaré inequalities are derived for invariant measures.

Results extend to interacting particle systems and McKean-Vlasov processes.

Abstract

We consider elliptic diffusion processes on $\mathbb R^d$. Assuming that the drift contracts distances outside a compact set, we prove that, at a sufficiently high temperature, the Markov semi-group associated to the process is a contraction of the $\mathcal W_2$ Wasserstein distance, which implies a Poincar\'e 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 Poincar\'e inequalities and quantitative long-time convergence for non-linear McKean-Vlasov type processes.