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

Abstract

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 Γ_2$-calculus)incompletegeneralityforinfinitesimallyHilbertianmetricmeasurespaces.Inparticular,weestablishthefullBochnerinequalityonsuchmetricmeasurespaces.Moreover,wededucenewcontractionboundsfortheheatflowonRiemannianmanifoldsandonmmsintermsofthe$L^2$-Wasserstein distance.

Additional results in Section 3.4 on dimension-dependent functional inequalities and Section 4.3 on applications to sharp spectral gap estimates. More detailed proof of the main result of Section 4.2

References in corpus (8)

• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Alexandrov meets Lott--Villani--Sturm
• Characterizations of Bounded Ricci Curvature on Smooth and NonSmooth Spaces
• Bochner-Weitzenboeck formula and Li-Yau estimates on Finsler manifolds
• Dimensional contraction via Markov transportation distance
• Geometry and Analysis of Dirichlet forms
• Non-branching geodesics and optimal maps in strong CD(K,{\infty})-spaces
• Ricci Bounds for Euclidean and Spherical Cones (revised/extended version)

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• Volume doubling, Poincaré inequality and Guassian heat kernel estimate for nonnegative curvature graphs
• $W$-entropy formulas and Langevin deformation of flows on Wasserstein space over Riemannian manifolds
• Convergence of Brownian motions on RCD(K,infty) spaces
• Failure of topological rigidity results for the measure contraction property
• $W_{1,+}$-interpolation of probability measures on graphs