A new class of transport distances between measures
A new class of transport distances between measures
复制标题
DOI：
10.1007/s00526-008-0182-5
复制
发表时间：
2009-02-01
影响因子：
2.1
通讯作者：
Savare, Giuseppe
中科院分区：
文献类型：
--
作者：
Dolbeault, Jean;Nazaret, Bruno;Savare, Giuseppe
我们在R-d中引入了一类新的非负Radon测度之间的距离。它们是以Benamou和Brenier(Numer Math 84：375-393,2000)提出的Kantorovich-Rubinstein-Wasserstein距离的动力学特征为模型的，并在Wasserstein距离和齐次W-Gamma(-1，p)-Sobolev距离之间提供了广泛的家族内插。从最优输运理论的观点来看，这些距离最小化了将给定的初始质量分布移动到最终构型的动态成本。与质量输运理论中经典设置的一个重要区别是，成本不仅取决于运动粒子的速度，还取决于相对于给定参考测量伽马的中间组态的密度。我们研究了这些新距离的拓扑和几何性质，并将它们与测度的弱收敛概念和著名的Kantorovich-Rubinstein-Wasserstein理论进行了比较。文中还给出了梯度流几何理论的一个可能应用实例。
We introduce a new class of distances between nonnegative Radon measures in R-d. They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier (Numer Math 84:375-393, 2000) and provide a wide family interpolating between the Wasserstein and the homogeneous W-gamma(-1,p)-Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure gamma. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich-Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given.