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

Ambrosio Luigi - Gigli Nicola - Savaré Giuseppe

year: 2011
abstract: 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.

The paper is available on the cvgmt preprint server.