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

The aim of the present paper is to bridge the gap between the Bakry-'{E}mery 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 Carr'{e} 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 $Gamma(f)le1$ admits a continuous representative. In such a class, we show that if ${{mathcal{E}}}$ satisfies a suitable weak form of the Bakry-'{E}mery 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-'{E}mery $mathrm {BE}(K,N)$ 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.