BAKRY-EMERY CURVATURE-DIMENSION CONDITION AND RIEMANNIAN RICCI CURVATURE BOUNDS

作者：Ambrsio, Luigi^*; Gigli, Nicola; Savare, Giuseppe

来源：Annals of Probability, 2015, 43(1): 339-404.

DOI：10.1214/14-AOP907

• Ricci curvature
• Barky-Emery condition
• metric measure space
• Dirichlet form
• Gamma calculus

The aim of the present paper is to bridge the gap between the Bakry-Emery 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 epsilon admitting a Carre du champ F in a Polish measure space (X, m) and a canonical distance de that induces the original topology of X. We first characterize the distinguished class of Riemannian Energy measure spaces, where epsilon coincides with the Cheeger energy induced by de and where every function f with Gamma (f) <= 1 admits a continuous representative. @@@ In such a class, we show that if epsilon satisfies a suitable weak form of the Bakry-Emery curvature dimension condition BE(K, infinity) then the metric measure space (X, d, m) satisfies the Riemannian Ricci curvature bound RCD(K, infinity) 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-Emery BE(K, N) condition (and thus the corresponding one for RCD(K, infinity) spaces without assuming non-branching) and the stability of BE(K, N) with respect to Sturm-Gromov-Hausdorff convergence.

• 出版日期2015-1