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This paper is devoted to a deeper understanding of the heat flow and to therefinement of calculus tools on metric measure spaces (X,d,m). Our main resultsare:  - A general study of the relations between the Hopf-Lax semigroup andHamilton-Jacobi equation in metric spaces (X,d).  - The equivalence of the heat flow in L^2(X,m) generated by a suitableDirichlet energy and the Wasserstein gradient flow of the relative entropyfunctional in the space of probability measures P(X).  - The proof of density in energy of Lipschitz functions in the Sobolev spaceW^{1,2}(X,d,m).  - A fine and very general analysis of the differentiability properties of alarge class of Kantorovich potentials, in connection with the optimal transportproblem.  Our results apply in particular to spaces satisfying Ricci curvature boundsin the sense of Lott & Villani [30] and Sturm [39,40], and require neither thedoubling property nor the validity of the local Poincar\'e inequality.

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