# On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces

@article{Erbar2013OnTE, title={On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces}, author={Matthias Erbar and Kazumasa Kuwada and Karl-Theodor Sturm}, journal={Inventiones mathematicae}, year={2013}, volume={201}, pages={993-1071} }

We prove the equivalence of the curvature-dimension bounds of Lott–Sturm–Villani (via entropy and optimal transport) and of Bakry–Émery (via energy and $$\Gamma _2$$Γ2-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 $$L^2$$L2-Wasserstein distance.

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