The aim of the present paper is to bridge the gap between the Bakry-Emery 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 Carre' du champ $\Gamma$ in a Polish measure space $(X,m)$ and a canonical distance $d_\mathcal E$ 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 $d_\mathcal E$ and where every function $f$ with $\Gamma (f)\leq 1$ admits a continuous representative. In such a class we show that if $E$ satisfies a suitable weak form of the Bakry-Emery curvature dimension condition $BE(K,\infty)$ then the metric measure space $(X,d,m)$ satisfies the Riemannian Ricci curvature bound $RCD(K,\infty)$ according to [5], thus showing the equivalence of the two notions. Two applications are then proved: the tensorization property for Riemannian Energy spaces satisfying the Bakry-Emery condition $BE(K,N)$ (and thus the corresponding one for $RCD(K,\infty)$ spaces without assuming nonbranching) and the stability of $BE(K,N)$ with respect to Sturm-Gromov-Hausdorff convergence.

Bakry-Emery curvature-dimension condition and Riemannian Ricci curvature bounds

Ambrosio, Luigi;Gigli, Nicola;Savaré, Giuseppe
2015-01-01

Abstract

The aim of the present paper is to bridge the gap between the Bakry-Emery 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 Carre' du champ $\Gamma$ in a Polish measure space $(X,m)$ and a canonical distance $d_\mathcal E$ 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 $d_\mathcal E$ and where every function $f$ with $\Gamma (f)\leq 1$ admits a continuous representative. In such a class we show that if $E$ satisfies a suitable weak form of the Bakry-Emery curvature dimension condition $BE(K,\infty)$ then the metric measure space $(X,d,m)$ satisfies the Riemannian Ricci curvature bound $RCD(K,\infty)$ according to [5], thus showing the equivalence of the two notions. Two applications are then proved: the tensorization property for Riemannian Energy spaces satisfying the Bakry-Emery condition $BE(K,N)$ (and thus the corresponding one for $RCD(K,\infty)$ spaces without assuming nonbranching) and the stability of $BE(K,N)$ with respect to Sturm-Gromov-Hausdorff convergence.
2015
43
1
339
404
https://arxiv.org/abs/1209.5786
Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/12638
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