Contractive coupling rates and curvature lower bounds for Markov chains

TL;DR

This paper demonstrates how contractive coupling rates can be used to establish curvature lower bounds and convexity properties for Markov chains, extending their application beyond Sobolev inequalities to geometric and entropic inequalities.

Contribution

It introduces the use of contractive coupling rates to prove curvature bounds and geodesic convexity for Markov chains, generalizing previous applications to Sobolev inequalities.

Findings

Established positive curvature in entropic and Bakry-Émery sense for specific Markov chains.

Connected coupling rates with coarse Ricci curvature and Wasserstein contraction.

Demonstrated exponential contraction of Wasserstein distance in examples.

Abstract

Contractive coupling rates have been recently introduced by Conforti as a tool to establish convex Sobolev inequalities (including modified log-Sobolev and Poincar\'{e} inequality) for some classes of Markov chains. In this work, we show how contractive coupling rates can also be used to prove stronger inequalities, in the form of curvature lower bounds for Markov chains and geodesic convexity of entropic functionals. We illustrate this in several examples discussed by Conforti, where in particular, after appropriately choosing a parameter function, we establish positive curvature in the entropic and (discrete) Bakry--\'{E}mery sense. In addition, we recall and give straightforward generalizations of some notions of coarse Ricci curvature, and we discuss some of their properties and relations with the concepts of couplings and coupling rates: as an application, we show exponential contraction of the $p$-Wasserstein distance for the heat flow in the aforementioned examples.