In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, global- to-local and local-to-global properties. In these spaces, that we call RCD spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry-E ́mery estimates and the Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger’s relaxed slope and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincare' and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones.

Metric measure spaces with Riemannian Ricci curvature bounded from below

Gigli, Nicola;
2014-01-01

Abstract

In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, global- to-local and local-to-global properties. In these spaces, that we call RCD spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry-E ́mery estimates and the Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger’s relaxed slope and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincare' and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones.
2014
163
7
1405
1490
https://arxiv.org/abs/1109.0222
Ambrosio, L.; Gigli, Nicola; Savaré, G.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/14217
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