Karl-Theodor Sturm (University of Bonn)
Mass transportation and nonlinear diffusions
Abstract.
We study nonlinear diffusions on a Riemannian manifold M as gradient flows with respect to suitable free energy functionals S on the space P(M) of probability measures over M. The space P(M), equipped with the L2-Wasserstein metric, will be regarded as an infinite dimensional manifold. Geodesics in P(M) are the paths of optimal mass transportation. Uniform convexity properties of S on P(M) will imply contraction and equilibration properties for the nonlinear diffusions on M, e.g. Talagrand inequality, logarithmic Sobolev inequality, exponential convergence to equilibrium. Of particular interest is the relative entropy with respect to the Riemannian volume. Its gradient flow turns out to be the heat equation. One of the main results is that a number K is a lower bound for the Hessian of the relative entropy on P(M) if and only if it is a lower bound for the Ricci curvature on M. This gives rise to a new definition of lower bounds for the Ricci curvature for general metric measure spaces. Also curvature-dimension conditions can be formulated. These conditions will be discussed in detail.

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