Diffusion, Optimal Transport and Ricci Curvature for Metric Measure Space

Lower Ricci curvature bounds play a crucial role in several deep geometric and functional inequalities in Riemannian geometry and diffusion processes. Bakry–Émery [8] introduced an elegant and powerful technique, based on commutator estimates for differential operators and so-called Γ-calculus, to derive many sharp results. Their curvature-dimension condition has been further developed by many authors, mainly in the framework of Markov diffusion modelled on weighted Riemannian manifolds, with relevant applications to infinite dimensional problems. A new synthetic approach relying on entropy and optimal transport has been more recently introduced by Lott, Sturm and Villani. It relies structurally on the notions of distance and measure and can therefore be used to extend the curvature-dimension condition to the general nonsmooth setting of metric measure spaces (X, d,m). Among its many beautiful properties, the synthetic approach is stable with respect to measured Gromov convergence. The equivalence of the two points of view can be directly proved in a smooth differential setting but it is a difficult task in a general metric framework, when explicit calculations in local charts are hard (if not impossible) to justify. We will try to give a brief and informal introduction to the two approaches and show how the Otto variational interpretation of the Fokker-Planck equation and the theory of metric gradient flows has provided a unifying point of view, which allows one to prove their equivalence for arbitrary metric measure spaces. As a byproduct, by combining Γ-calculus and optimal transport techniques, an impressive list of deep results in Riemannian geometry and smooth diffusion have a natural counterpart in the nonsmooth metric measure framework.