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Abstract |
This talk is based on a joint work with M. Erbar and K.-T. Sturm (Bonn). In this joint work, the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) is shown to be equivalent to Bakry–\'Emery's one (via energy and $\Gamma_2$-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. In this talk, I will explain some of other equivalent conditions and implications between them. It includes the introduction of two conditions: an alternative curvature-dimension bound via optimal transport using the relative entropy, which we call the entropic curvature-dimension condition, and a space-time Lipschitz-type bounds for the heat flow in terms of the $L^2$-Wasserstein distance. These conditions are new even on smooth spaces such as Riemannian manifolds. |
