Transport-entropy inequalities and curvature in discrete-space markov chains

Abstract

Let G=(Ω, E) be a graph and let d be the graph distance. Consider a discrete-time Markov chain {Z t } on Ω whose kernel p satisfies p(x, y) > 0 ⇒ {x, y} ε E for every x, yεΩ. In words, transitions only occur between neighboring points of the graph. Suppose further that (Ω, p, d) has coarse Ricci curvature at least 1/α in the sense of Ollivier: For all x, yεΩ, it holds that (formula presented) where W1 denotes the Wasserstein 1-distance. In this note, we derive a transport-entropy inequality: For any measure µ on Ω, it holds that (formula presented) where π denotes the stationary measure of {Z t } and D(.∥.) is the relative entropy. Peres and Tetali have conjectured a stronger consequence of coarse Ricci curvature, that a modified log-Sobolev inequality (MLSI) should hold, in analogy with the setting of Markov diffusions. We discuss how our approach suggests a natural attack on the MLSI conjecture.