Ricci curvature of metric spaces
Ricci curvature of metric spaces
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DOI：
10.1016/j.crma.2007.10.041
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发表时间：
2007-12-01
影响因子：
0.8
通讯作者：
Ollivier, Yann
中科院分区：
文献类型：
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作者：
Ollivier, Yann
我们在带有测度或随机漫步的度量空间中定义了里奇曲率的概念。为此，我们使用随机游走的局部收缩系数作用于具有运输距离的概率测度空间。这个概念允许推广与正里奇曲率相关的几个经典定理，例如谱间隙界（Lichnerowicz定理），高斯测度浓度（Levy-Gromov定理），对数Sobolev不等式（Bakry-Emery理论的结果）或Bonnet-Myers定理。该定义与Bakry-Emery理论兼容，具有鲁棒性，易于在图等具体实例中实现。
We define a notion of Ricci curvature in metric spaces equipped with a measure or a random walk. For this we use a local contraction coefficient of the random walk acting on the space of probability measures equipped with a transportation distance. This notions allows to generalize several classical theorems associated with positive Ricci curvature, such as a spectral gap bound (Lichnerowicz theorem), Gaussian concentration of measure (Levy-Gromov theorem), logarithmic Sobolev inequalities (a result of Bakry-Emery theory) or the Bonnet-Myers theorem. The definition is compatible with Bakry-Emery theory, and is robust and very easy to implement in concrete examples such as graphs.