This is a condensed form of the author's essay, which can be found at [arXiv:1105.2883]. We prove that the entropic measure constructed by von Renesse-Sturm over Wasserstein space on the unit interval (probability measures on the unit interval equipped with the 2-Wasserstein metric) does not admit generalized Ricci lower bounds in the sense of Lott-Villani-Sturm. We discuss why this is surprising, considering various heuristic arguments.
On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner's Inequality on Metric Measure Spaces
我们证明了Lott-Sturm-Villani（通过熵和最优输运）和Bakry-Emery（通过能量和Γ 2 $-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the $ L^2$-Wasserstein距离）的曲率维数界的等价性.
Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
本文证明了RCD（K，\infty）度量测度空间（X，d，m）中的线性热流关于每个L^p-Kantorovich-Rubinstein-Wasserstein距离都满足收缩性质.特别地，我们得到了两个基本解之间的最优W_ \infty -耦合关于初始点距离的精确估计.这个结果是（X，d，m）中正则Cheeger-Dirichlet型的RCD（K，\infty）Ricci下界与相应的Bakry-Émery条件之间等价的结果.关键的工具是扩展到非光滑的度量措施设置的巴克里的论点，这使得能够提高之间的交换估计的马尔可夫半群和Carré du Champ相关联的狄利克雷形式。这种扩展是基于一个新的先验估计和一个容量参数的定期和紧狄利克雷形式是独立的利益。
