Ricci curvature of Markov chains on metric spaces

Yann Ollivier

Published 2007-01-30, updated 2007-07-30Version 4

We 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--\'Emery Ricci curvature for Brownian motion with a drift on a Riemannian manifold. Positive Ricci curvature is easily shown to imply a spectral gap, a L\'evy--Gromov-like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. These bounds are sharp in several interesting examples.