Entropy-Wasserstein regularization, defective local concentration and a cutoff criterion beyond non-negative curvature

TL;DR

This paper introduces a relaxed form of curvature for Markov processes, establishing local concentration and regularization effects, and applies these to analyze cutoff phenomena in negatively curved settings.

Contribution

It develops a weaker Wasserstein curvature notion and proves local concentration and entropy-transport regularization under this framework.

Findings

Establishes defective Talagrand inequalities for Markov kernels.

Demonstrates entropy-transport regularization effects.

Provides criteria for cutoff phenomena in negatively curved Markov processes.

Abstract

Notions of positive curvature have been shown to imply many remarkable properties for Markov processes, in terms, e.g., of regularization effects, functional inequalities, mixing time bounds and, more recently, the cutoff phenomenon. In this work, we are interested in a relaxed variant of Ollivier's coarse Ricci curvature, where a Markov kernel $P$ satisfies only a weaker Wasserstein bound $W_p(\mu P, \nu P) \leq K W_p(\mu,\nu)+M$ for constants $M\ge 0, K\in [0,1], p \ge 1$. Under appropriate additional assumptions on the one-step transition measures $\delta_x P$, we establish (i) a form of local concentration, given by a defective Talagrand inequality, and (ii) an entropy-transport regularization effect. We consider as illustrative examples the Langevin dynamics and the Proximal Sampler when the target measure is a log-Lipschitz perturbation of a log-concave measure. As an application of the above results, we derive criteria for the occurrence of the cutoff phenomenon in some negatively curved settings.