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

Abstract

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.