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February 2015 Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds
Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré
Ann. Probab. 43(1): 339-404 (February 2015). DOI: 10.1214/14-AOP907

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.

Citation

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Luigi Ambrosio. Nicola Gigli. Giuseppe Savaré. "Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds." Ann. Probab. 43 (1) 339 - 404, February 2015. https://doi.org/10.1214/14-AOP907

Information

Published: February 2015
First available in Project Euclid: 12 November 2014

zbMATH: 1302.83023
MathSciNet: MR3298475
Digital Object Identifier: 10.1214/14-AOP907

Subjects:
Primary: 47D07, 49Q20
Secondary: 30L99

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 1 • February 2015
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