Bakry-\'Emery 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 $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }(f)\le 1$ admits a continuous representative. In such a class, we show that if $\mathcal{E}$ satisfies a suitable weak form of the Bakry-'{E}mery curvature dimension condition $\mathrm{B}\mathrm{E}(K,\mathrm{\infty })$ then the metric measure space $(X,\mathsf{d},\mathfrak{m})$ satisfies the Riemannian Ricci curvature bound $\mathrm{R}\mathrm{C}\mathrm{D}(K,\mathrm{\infty })$ 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 $\mathrm{B}\mathrm{E}(K,N)$ condition (and thus the corresponding one for $\mathrm{R}\mathrm{C}\mathrm{D}(K,\mathrm{\infty })$ spaces without assuming nonbranching) and the stability of $\mathrm{B}\mathrm{E}(K,N)$ with respect to Sturm-Gromov-Hausdorff convergence.