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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Abstract

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

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

Dates
First available in Project Euclid: 12 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1415801559

Digital Object Identifier
doi:10.1214/14-AOP907

Mathematical Reviews number (MathSciNet)
MR3298475

Zentralblatt MATH identifier
1302.83023

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aop/1415801559


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