Non-negative Ollivier curvature on graphs, reverse Poincaré inequality, Buser inequality, Liouville property, Harnack inequality and eigenvalue estimates

Abstract

We prove that for combinatorial graphs with non-negative Ollivier curvature, one has‖Ptμ−Ptν‖1≤W1(μ,ν)t for all probability measures μ,ν where Pt is the heat semigroup and W1 is the ℓ1-Wasserstein distance. This turns out to be an equivalent formulation of a version of reverse Poincaré inequality. Furthermore, this estimate allows us to prove Buser inequality, Liouville property and the eigenvalue estimate λ1≥log⁡(2)/diam2.