BAKRY–ÉMERY CURVATURE-DIMENSION CONDITION AND RIEMANNIAN RICCI CURVATURE BOUNDS

• Published in: The Annals of applied probability 2015-01, Vol.43 (1), p.339-404
• Main Authors: Ambrosio, Luigi, Gigli, Nicola, Savaré, Giuseppe
• Format: Article
• Language: English
• Subjects: 30L99; 47D07; 49Q20; Analysis of PDEs; Barky–Émery condition; Dirichlet form; Functional Analysis; Gamma calculus; Mathematics; Metric Geometry; metric measure space; Probability; Ricci curvature
• ISSN: 0091-1798; 1050-5164; 2168-8737; 2168-894X
• DOI: 10.1214/14-AOP907

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 ε admitting a Carré du champ Γ in a Polish measure space (X, m) and a canonical distance dε that induces the original topology of X. We first characterize the distinguished class of Riemannian Energy measure spaces, where ε coincides with the Cheeger energy induced by dε and where every function f with Γ(f) ≤ 1 admits a continuous representative. In such a class, we show that if ε satisfies a suitable weak form of of Bakry–Émery curvature dimension condition BE(K, ∞) then the metric measure space (X, d, m) satisfies the Riemannian Ricci curvature bound RCD(K, ∞) 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 BE(K, N) condition (and thus the corresponding one for RCD(K, ∞) spaces without assuming non-branching) and the stability of BE(K, N) with respect to Sturm–Gromov–Hausdorff convergence.