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Transport inequalities, gradient estimates, entropy and Ricci curvature. (English) Zbl 1078.53028

Authors’ abstract: 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 semigroup.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J65 Diffusion processes and stochastic analysis on manifolds
35K05 Heat equation
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