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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