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*Submitted Paper*

**Inserted:** 30 jul 2011

**Last Updated:** 7 may 2013

**Year:** 2011

**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 $RCD(K,\infty)$ 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-Emery estimates and the $L^\infty-{\rm Lip}$ 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 PoincarÃ© and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones.

**Tags:**
GeMeThNES

**Keywords:**
Optimal transport, entropy, Ricci curvature, Heat Flow

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