测地线度量空间中的分歧最优迁移
Ramified optimal transportation in geodesic metric spaces
摘要：
最优输运路径可以被看作是在合适的度量族下的概率测度空间中的测地线。这种测地线可以在许多应用中表现出树形分支结构，例如树木、血管、排水和灌溉系统。本文将概率测度之间分歧最优运输的研究从欧几里得空间推广到测地线度量空间。我们调查的存在性以及最优输运路径的各种性质下的度量，如完整性，加倍，或曲率上界的行为。在完备测地度量空间上引入了概率测度的迁移维数，并证明了概率测度的迁移维数上界为Minkowski维数，下界为测度的豪斯多夫维数.此外，我们还在概率测度空间上引入了一个度量，称为“维数距离”。这个度量给了传输维一个几何意义：关于这个度量，一个概率测度的传输维等于从它到任何有限原子概率测度的距离。
英文摘要：
An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the study of ramified optimal transportation between probability measures from Euclidean spaces to a geodesic metric space. We investigate the existence as well as the behavior of optimal transport paths under various properties of the metric such as completeness, doubling, or curvature upper boundedness. We also introduce the transport dimension of a probability measure on a complete geodesic metric space, and show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called "the dimensional distance", on the space of probability measures. This metric gives a geometric meaning to the transport dimension: with respect to this metric, the transport dimension of a probability measure equals to the distance from it to any finite atomic probability measure.
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