Inserted: 10 jun 2011
Last Updated: 17 nov 2012
This and a companion forthcoming paper are devoted to a deeper understanding of the heat flow in metric measure spaces $(X,d,m)$. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott & Villani and Sturm. Indeed, the development of a ``calculus'' in this class of spaces is one of our motivations. In this paper the main goals are:
(i) The proof of equivalence of the heat flow in $L^2$ generated by a suitable Dirichlet energy and the Wasserstein gradient flow in the space of probability measuress of the relative entropy functional w.r.t. $m$.
(ii) The equivalence of two weak notions of modulus of the gradient: the first one (inspired by Cheeger), that we call relaxed gradient, is defined by $L^2(X,\mm)$-relaxation of the pointwise Lipschitz constant in the class of Lipschitz functions; the second one (inspired by Shanmugalingam), that we call weak upper gradient, is based on the validity of the fundamental theorem of calculus along almost all curves. These two notions of gradient will be compared and identified under very mild assumptions on $(X,d,m)$ which include all finite measures. Under additional assumptions, fulfilled in $LSV$ spaces, these derivatives will be identified with a third object, namely the energy density appearing in the so-called Fisher information functional, representing the energy dissipation rate of entropy w.r.t. the Wasserstein distance.
(iii) A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem.
Keywords: Optimal transport, entropy, Ricci curvature, Heat Flow