Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds

Abstract

The aim of the present paper is to bridge the gap between the Bakry-Émery and the Lott-Sturm-Villani approaches to provide synthetic and abstract notions of lower Ricci curvature bounds. We start from a strongly local Dirichlet form $\mathcal{E}$ admitting a Carré du champ $\Gamma$ in a Polish measure space $(X,\mathfrak{m})$ and a canonical distance ${\mathsf{d}}_{\mathcal{E}}$ that induces the original topology of $X$. We first characterize the distinguished class of Riemannian Energy measure spaces, where $\mathcal{E}$ coincides with the Cheeger energy induced by ${\mathsf{d}}_{\mathcal{E}}$ and where every function $f$ with $\Gamma (f)\le 1$ admits a continuous representative. In such a class, we show that if $\mathcal{E}$ satisfies a suitable weak form of the Bakry-Émery curvature dimension condition $\mathrm{BE}(K,\infty )$ then the metric measure space $(X,\mathsf{d},\mathfrak{m})$ satisfies the Riemannian Ricci curvature bound $\mathrm{RCD}(K,\infty )$ according to [Duke Math. J. 163 (2014) 1405-1490], thus showing the equivalence of the two notions. Two applications are then proved: the tensorization property for Riemannian Energy spaces satisfying the Bakry-Émery $\mathrm{BE}(K,N)$ condition (and thus the corresponding one for $\mathrm{RCD}(K,\infty )$ spaces without assuming nonbranching) and the stability of $\mathrm{BE}(K,N)$ with respect to Sturm-Gromov-Hausdorff convergence.

Published in at http://dx.doi.org/10.1214/14-AOP907 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

Cited by in corpus (88)

• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures
• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
• The splitting theorem in non-smooth context
• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
• Well posedness of Lagrangian flows and continuity equations in metric measure spaces
• Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds
• Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows
• Li-Yau and Harnack type inequalities in $RC{D}^{\ast }(K,N)$ metric measure spaces
• New formulas for the Laplacian of distance functions and applications
• Obata's rigidity theorem for metric measure spaces
• Polya-Szego inequality and Dirichlet $p$-spectral gap for non-smooth spaces with Ricci curvature bounded below
• A review of Lorentzian synthetic theory of timelike Ricci curvature bounds
• Harmonic functions on metric measure spaces
• Characterizations of sets of finite perimeter using heat kernels in metric spaces
• The Heat Flow on Metric Random Walk Spaces
• Dimensional contraction via Markov transportation distance
• A Noncommutative Transport Metric and Symmetric Quantum Markov Semigroups as Gradient Flows of the Entropy
• Sobolev spaces on warped products
• Measure rigidity of Ricci curvature lower bounds
• Lecture notes on differential calculus on RCD spaces
• Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries
• Rademacher-type Theorems and Sobolev-to-Lipschitz Properties for Strongly Local Dirichlet Spaces
• Hamilton's Gradient Estimates and A Monotonicity Formula for Heat Flows on Metric Measure Spaces
• Gaussian-type Isoperimetric Inequalities in $RCD(K,\infty )$ probability spaces for positive $K$
• Rectifiability of the reduced boundary for sets of finite perimeter over RCD$(K,N)$ spaces
• Second order differentiation formula on compact $RC{D}^{\ast }(K,N)$ spaces
• Maximal volume entropy rigidity for ${\mathsf{RCD}}^{\ast }(-(N-1),N)$ spaces
• Local spectral convergence in $RC{D}^{\ast }(K,N)$ spaces
• Rigidity for the spectral gap on $RCD(K,\infty )$-spaces
• Weyl's law on $RC{D}^{\ast }(K,N)$ metric measure spaces
• Functional inequalities for the heat flow on time-dependent metric measure spaces
• An overview on the proof of the splitting theorem in non-smooth context
• Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows
• Optimal transport, gradient estimates, and pathwise Brownian coupling on spaces with variable Ricci bounds
• Recognizing the flat torus among $RC{D}^{\ast }(0,N)$ spaces via the study of the first cohomology group
• Heat Kernel Bounds on Metric Measure Spaces and Some Applications
• Super-Ricci flows and improved gradient and transport estimates
• Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds
• Differential forms on Dirichlet spaces and Bakry-Émery estimates on metric graphs
• Characterizations of monotonicity of vector fields on metric measure spaces
• Geometric properties of Dirichlet forms under order isomorphisms
• Sharp Heat Kernel Bounds and Entropy in Metric Measure Spaces
• Bakry-Émery curvature and model spaces in sub-Riemannian geometry
• From volume cone to metric cone in the nonsmooth setting
• Sampling and Optimization on Convex Sets in Riemannian Manifolds of Non-Negative Curvature
• Sobolev-to-Lipschitz Property on QCD-spaces and Applications
• Bakry-Émery conditions on almost smooth metric measure spaces
• Weakly non-collapsed RCD spaces are strongly non-collapsed
• Coarse Ricci curvature of hypergraphs and its generalization
• Isometric immersions of RCD$(K,N)$ spaces via heat kernels
• The metric measure boundary of spaces with Ricci curvature bounded below
• Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions
• Convergence of Non-Symmetric Diffusion Processes on RCD spaces
• The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces
• Moduli spaces of compact RCD(0,N)-structures
• Limits of manifolds with a Kato bound on the Ricci curvature
• Harnack inequalities on totally geodesic foliations with transverse Ricci flow
• Weighted Littlewood--Paley inequalities for heat flows in $\RCD$ spaces
• Harmonic mappings between singular metric spaces
• Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces
• Curvature-dimension conditions for diffusions under time change
• Stability of Kac regularity under domination of quadratic forms
• Quantum optimal transport for approximately finite-dimensional $C^{\ast }$-algebras
• Ricci tensor on smooth metric measure space with boundary
• Sharp Spectral Gaps on Metric Measure Spaces
• Generalized Bakry-Émery curvature condition and equivalent entropic inequalities in groups
• Hölder continuity of tangent cones in RCD(K,N) spaces and applications to non-branching
• An entropy generation formula on $RCD(K,\infty )$ spaces
• Convergence of Brownian Motions on Metric Measure Spaces Under Riemannian Curvature-Dimension Conditions
• The normal contraction property for non-bilinear Dirichlet forms
• Curvature bound of Dyson Brownian Motion
• Coarse Ricci curvature as a function on $M\times M$
• Cheng maximal diameter theorem for hypergraphs
• Littlewood--Paley--Stein inequalities on $\text{RCD}(K,\infty )$ spaces
• Noncompact RCD(0,N) spaces with linear volume growth
• Characterization of equality in Zhong-Yang type (sharp) spectral gap estimates for metric measure spaces
• A characterization of the rate of change of $\Phi$-entropy via an integral form curvature-dimension condition
• Bakry-Émery Ricci curvature of doubly warped product of weighted spaces
• Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below
• Rigidity of some functional inequalities on RCD spaces
• Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian
• Some functional inequalities on non-reversible Finsler manifolds
• Curvature-dimension conditions for symmetric quantum Markov semigroups
• Mosco convergence of gradient forms with non-convex interaction potential
• Persistence of Rademacher-type and Sobolev-to-Lipschitz properties
• The stochastic value function on metric measure spaces
• Sharp gradient estimate for heat kernels on $RC{D}^{\ast }(K,N)$ metric measure spaces