Contractive coupling rates and curvature lower bounds for Markov chains

Abstract

Contractive coupling rates have been recently introduced by Conforti as a tool to establish convex Sobolev inequalities (including modified log-Sobolev and Poincaré 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--Émery 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.

49 pages, final version, accepted in AAP

References in corpus (5)

• Ricci curvature of finite Markov chains via convexity of the entropy
• Eulerian calculus for the displacement convexity in the Wasserstein distance
• Entropic Ricci curvature bounds for discrete interacting systems
• Coarse Ricci curvature for continuous-time Markov processes
• Entropy decay for interacting systems via the Bochner--Bakry--Emery approach