Diffusion, optimal transport and Ricci curvature

  • Giuseppe Savaré

    Università di Pavia, Italy
Diffusion, optimal transport and Ricci curvature cover
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Abstract

Starting from the pioneering paper of Otto-Villani [73], the link between Optimal Transport and Ricci curvature in smooth Riemannian geometry has been deeply studied [35, 86]. Among the various functional and analytic applications, the point of view of Optimal Transport has played a crucial role in the Lott–Sturm–Villani [69, 84, 85, 87] 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,d,m)(X,\mathrm 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 [21] approach based on Markov semigroups, diffusion operators and Γ\Gamma-calculus for strongly local Dirichlet forms [22].

We will discuss a series of recent contributions [5, 7–9, 12, 38] showing the link of both the approaches with the metric-variational theory of gradient flows [6] and diffusion equations. As a byproduct, when the Cheeger energy on „(X,d,m)(X,\mathrm d,\mathfrak m) is quadratic (or, equivalently, the Sobolev space W1,2(X,d,m)W^{1,2}(X,\mathrm 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.