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Contractive coupling rates have been recently introduced by Conforti as atool to establish convex Sobolev inequalities (including modified log-Sobolevand Poincar\'{e} inequality) for some classes of Markov chains. In this work,we show how contractive coupling rates can also be used to prove strongerinequalities, in the form of curvature lower bounds for Markov chains andgeodesic convexity of entropic functionals. We illustrate this in severalexamples discussed by Conforti, where in particular, after appropriatelychoosing a parameter function, we establish positive curvature in the entropicand (discrete) Bakry--\'{E}mery sense. In addition, we recall and givestraightforward generalizations of some notions of coarse Ricci curvature, andwe discuss some of their properties and relations with the concepts ofcouplings and coupling rates: as an application, we show exponentialcontraction of the $p$-Wasserstein distance for the heat flow in theaforementioned examples.

Contractive coupling rates and curvature lower bounds for Markov chains