SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2849-2873, January 2020.
We suggest a new way of defining optimal transport of positive-semidefinite matrix-valued measures. It is inspired by a recent rendering of the incompressible Euler equations and related conservative systems as concave maximization problems. The main object of our attention is the Kantorovich--Bures metric space, which is a matricial analogue of the Wasserstein and Hellinger--Kantorovich metric spaces. We establish some topological, metric, and geometric properties of this space, which includes the existence of the optimal transportation path.

中文翻译：

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关于矩阵值测度的最优输运

SIAM数学分析杂志，第52卷，第3期，第2849-2873页，2020年1月。
我们提出了一种定义正半定矩阵值测度的最优输运的新方法。它的灵感来自于最近将不可压缩的Euler方程和相关的保守系统渲染为凹最大化问题。我们关注的主要对象是Kantorovich-Bures度量空间，它是Wasserstein和Hellinger-Kantorovich度量空间的矩阵类似形式。我们建立了该空间的一些拓扑，度量和几何属性，其中包括最佳运输路径的存在。