Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds

Abstract

We present some new results concerning well-posedness of gradient flows generated by λ-convex functionals in a wide class of metric spaces, including Alexandrov spaces satisfying a lower curvature bound and the corresponding L 2 -Wasserstein spaces. Applications to the gradient flow of Entropy functionals in metric-measure spaces with Ricci curvature bounded from below and to the corresponding diffusion semigroup are also considered. These results have been announced during the workshop on “Optimal Transport: theory and applications” held in Pisa, November 2006. To cite this article: G. Savaré, C. R. Acad. Sci. Paris, Ser. I 345 (2007).