A variational approach to gradient flows in metric-measure spaces

Giuseppe Savaré (Università di Pavia, Italy)

Combining optimal transportation techniques and new convergence estimates for the variational formulation of the implicit Euler scheme, the generation of an heat/Fokker-Planck diffusion semigroup can be obtained in a wide class of metric-measure spaces satisfying a lower Ricci curvature bound, in the general framework recently developed by Sturm and Lott-Villani.

Besides the possibility of an "intrinsic" construction, involving the Wasserstein distance and the entropy functional as in Otto's fundamental contribution, the metric formulation enjoys nice stability properties with respect to measured Gromov-Hausdorff convergence, covering also the case of Gaussian and log-concave measures in infinite-dimensional Hilbert spaces.

In this talk we will present some aspects of this theory.