METRIC MEASURE SPACES WITH RIEMANNIAN RICCI CURVATURE BOUNDED FROM BELOW
METRIC MEASURE SPACES WITH RIEMANNIAN RICCI CURVATURE BOUNDED FROM BELOW
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DOI：
10.1215/00127094-2681605
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发表时间：
2014-05-15
影响因子：
2.5
通讯作者：
Savare, Giuseppe
中科院分区：
文献类型：
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作者：
Ambrosio, Luigi;Gigli, Nicola;Savare, Giuseppe
本文给出了度量度量空间(X，d，m)的Riemannian Ricci界的一个综合概念，它在测量的Gromov-Hausdorff收敛下是稳定的，并排除了Finsler几何。它可以通过加强Lott、Sturm和Villani测地线凸性条件来给出，该条件与热流的线性耦合。除了稳定性，它还具有相同的张化性、全局到局部和局部到全局的性质。在这些称为RCD(K，无穷)的空间中，我们证明了热流(可以等价地刻画为与Dirichlet形式相关的流，或熵的Wasserstein梯度流)满足Wasserstein压缩估计和几个正则性，特别是Bakry-Emery估计和L-无穷大-Lip-Feller正则化。我们还证明了由Dirichlet形式诱导的距离与d重合，局部能量度量具有由Cheeger松弛斜率的平方给出的密度，因此，基础布朗运动有连续的路径。所有这些结果都是独立于关于度量结构的Poincare和加倍假设而得到的，因此也适用于不是局部紧的空间，就像无限维空间一样。
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, which we call RCD(K, infinity) 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-infinity-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 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.