Calculus and heat flow in metric measure spaces and applications to   spaces with Ricci bounds from below
  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. 
Luigi Ambrosio, Nicola Gigli, Giuseppe Savar\'e
math.MG math.AP math.FA