Reflection couplings and contraction rates for diffusions

TL;DR

This paper introduces a novel approach to analyze the contractivity of diffusion semigroups using concave Wasserstein-like distances, achieving near-optimal rates in cases where standard Wasserstein distances fail, with applications to Langevin diffusions and interacting particle systems.

Contribution

It develops explicit distance functions that enable contractivity results for diffusions in scenarios where traditional Wasserstein metrics are ineffective.

Findings

Achieves near-optimal contraction rates for certain diffusions.

Extends contractivity analysis to non-convex potentials and interacting systems.

Provides new tools for studying convergence of complex stochastic processes.

Abstract

We consider contractivity for diffusion semigroups w.r.t. Kantorovich ($L^1$ Wasserstein) distances based on appropriately chosen concave functions. These distances are inbetween total variation and usual Wasserstein distances. It is shown that by appropriate explicit choices of the underlying distance, contractivity with rates of close to optimal order can be obtained in several fundamental classes of examples where contractivity w.r.t. standard Wasserstein distances fails. Applications include overdamped Langevin diffusions with locally non-convex potentials, products of these processes, and systems of weakly interacting diffusions, both of mean-field and nearest neighbour type.