The Annals of Probability

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

Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré

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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 $\mathcal{E}$ admitting a Carré 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–Émery 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–Émery $\mathrm{BE} (K,N)$ 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.

Article information

Ann. Probab., Volume 43, Number 1 (2015), 339-404.

First available in Project Euclid: 12 November 2014

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Zentralblatt MATH identifier

Primary: 49Q20: Variational problems in a geometric measure-theoretic setting 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}
Secondary: 30L99: None of the above, but in this section

Ricci curvature Barky–Émery condition metric measure space Dirichlet form Gamma calculus


Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe. Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab. 43 (2015), no. 1, 339--404. doi:10.1214/14-AOP907.

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