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arXiv · 1303.4382

On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner's Inequality on Metric Measure Spaces

Matthias Erbar·Kazumasa Kuwada·Karl-Theodor Sturm
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Abstract

We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry--\'Emery (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. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the $L^2$-Wasserstein distance.

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Matthias Erbar, Kazumasa Kuwada, Karl-Theodor Sturm. 2013-03-18. On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner's Inequality on Metric Measure Spaces. https://arxiv.org/abs/1303.4382

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