Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

Luigi Ambrosio (1) , Nicola Gigli (2) , Giuseppe Savaré (3)

1 SNS - Scuola Normale Superiore di Pisa
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 Dipartimento di matematica F. Casorati

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X,d,m). Our main results are: - A general study of the relations between the Hopf-Lax semigroup and Hamilton-Jacobi equation in metric spaces (X,d). - The equivalence of the heat flow in L^2(X,m) generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional in the space of probability measures P(X). - The proof of density in energy of Lipschitz functions in the Sobolev space W^{1,2}(X,d,m). - A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott & Villani [30] and Sturm [39,40], and require neither the doubling property nor the validity of the local Poincaré inequality.

Domaines: Géométrie métrique [math.MG]; Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré. Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. 2012. ⟨hal-00769378⟩