Curvature and transport inequalities for Markov chains in discrete spaces

Abstract

We study various transport-information inequalities under three different notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and Emery (In Seminaire de Probabilites, XIX, 1983/84 (1985) 177–206 Springer), the exponential curvature-dimension condition of Bauer et al. (Li-Yau Inequality on Graphs (2013)) and the coarse Ricci curvature of Ollivier (J. Funct. Anal. 256 (2009) 810–864). We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an $L_{1}$ transport-information inequality holds; while under an exponential curvature-dimension condition, some weak-transport information inequalities hold. As an application, we establish a Bonnet–Myers theorem under the curvature-dimension condition $\operatorname{CD}(\kappa,\infty)$ of Bakry and Emery (In Seminaire de Probabilites, XIX, 1983/84 (1985) 177–206 Springer).