Diffusion, optimal transport and Ricci curvature
Summary: Starting from the pioneering paper of Otto-Villani, the link between Optimal Transport and Ricci curvature in smooth Riemannian geometry has been deeply studied. Among the various functional and analytic applications, the point of view of Optimal Transport has played a crucial role in the Lott-Sturm-Villani formulation of a ``synthetic'' notion of lower Ricci curvature bound, which has been extended from the realm of smooth Riemannian manifold to the general framework of metric measure spaces \((X,\mathsf d,\mathfrak m)\), i.e., (separable, complete) metric spaces endowed with a finite or locally finite Borel measure \(\mathfrak m\). Lower Ricci curvature bounds can also be captured by the celebrated Bakry-Émery approach based on Markov semigroups; diffusion operators and \(\Gamma\)-calculus for strongly local Dirichlet forms. We will discuss a series of recent contributions showing the link of both the approaches with the metric-variational theory of gradient flows [\textit{L. Ambrosio} et al., Gradient flows in metric spaces and in the space of probability measures. 2nd ed. Basel: Birkhäuser (2008; Zbl 1145.35001)] and diffusion equations. As a byproduct, when the Cheeger energy on \((X,\mathsf d,\mathfrak m)\) is quadratic (or, equivalently, the Sobolev space \(W^{1,2}(X,\mathsf d,\mathfrak m)\) is Hilbertian), we will show that the two approaches lead to essentially equivalent definitions and to a nice geometric framework suitable for deep analytic results. For the entire collection see [Zbl 1396.00017].
scientific article; zbMATH DE number 6979391
scientific article