Transport-Entropy Inequalities and Curvature in Discrete-Space Markov Chains

Let (G = (Omega,E)) be a graph and let d be the graph distance. Consider a discrete-time Markov chain {Z t } on (Omega) whose kernel p satisfies p(x, y) u003e 0 ⇒ {x, y} ∈ E for every (x,y Omega). In words, transitions only occur between neighboring points of the graph. Suppose further that ((Omega,p,d)) has coarse Ricci curvature at least 1∕α in the sense of Ollivier: For all (x,y Omega), it holds that (W_{1}(Z_{1}mid {Z_{0} = x},Z_{1}mid {Z_{0} = y}) leq left (1 -frac{1} {alpha } right )d(x,y),) where W1 denotes the Wasserstein 1-distance.