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

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 E admitting a Carré du champ Γ 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 E coincides with the Cheeger energy induced by dE and where every function f with Γ(f) ≤ 1 admits a continuous representative. In such a class we show that if E satisfies a suitable weak form of the 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 [5], thus showing the equivalence of the two notions. Two applications are then proved: the tensorization property for Riemannian Energy spaces satisfying the Bakry-Émery condition BE(K,N) (and thus the corresponding one for RCD(K,∞) spaces without assuming nonbranching) and the stability of BE(K,N) with respect to Sturm-Gromov-Hausdorff convergence.