TY - JOUR
T1 - On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
AU - Erbar, Matthias
AU - Kuwada, Kazumasa
AU - Sturm, Karl Theodor
N1 - Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg.
PY - 2015/9/22
Y1 - 2015/9/22
N2 - 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.
AB - 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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U2 - 10.1007/s00222-014-0563-7
DO - 10.1007/s00222-014-0563-7
M3 - Article
AN - SCOPUS:84939772144
VL - 201
SP - 993
EP - 1071
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
SN - 0020-9910
IS - 3
ER -