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

TL;DR

This paper advances the understanding of heat flow and calculus in metric measure spaces, establishing key equivalences and properties that apply to spaces with Ricci curvature bounds without requiring doubling or Poincaré inequalities.

Contribution

It provides new results on the relation between heat flow and optimal transport, and analyzes differentiability of Kantorovich potentials in general metric measure spaces.

Findings

Equivalence of heat flow and Wasserstein gradient flow of entropy

Density of Lipschitz functions in Sobolev space

Differentiability properties of Kantorovich potentials

Abstract

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 Poincar\'e inequality.