  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.
