Transport inequalities, gradient estimates, entropy, and Ricci curvature

We present various characterizations of uniform lower bounds for the Ricci curvature of a smooth Riemannian manifold M in terms of convexity properties of the entropy (considered as a function on the space of probability measures on M) as well as in terms of transportation inequalities for volume measures, heat kernels, and Brownian motions and in terms of gradient estimates for the heat semi-group. (c) 2004 Wiley Periodicals, Inc.