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Article overview
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Metric measure spaces with Riemannian Ricci curvature bounded from below | Luigi Ambrosio
; Nicola Gigli
; Giuseppe Savaré
; | Date: |
1 Sep 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-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 Poincar’e and doubling assumptions on the metric measure
structure and therefore apply also to spaces which are not locally compact, as
the infinite-dimensional ones | Source: | arXiv, 1109.0222 | Services: | Forum | Review | PDF | Favorites |
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