2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/135179We define the Ricci curvature of Markov chains on metric spaces as a local contraction coefficient of the random walk acting on the space of probability measures equipped with a Wasserstein transportation distance. For Brownian motion on a Riemannian manifold this gives back the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein--Uhlenbeck process. Moreover this generalization is consistent with the Bakry--Émery Ricci curvature for Brownian motion with a drift on a Riemannian manifold. Positive Ricci curvature is easily shown to imply a spectral gap, a Lévy--Gromov-like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. These bounds are sharp in several interesting examples.ProbabilityMetric Geometry51F99, 53B21, 60B99text