论文标题
不平衡度量空间之间的熵传输距离
Entropy-Transport distances between unbalanced metric measure spaces
论文作者
论文摘要
受到熵传输问题的最新理论的启发，以及$ \ mathbf {d} $ - 在归一化度量测量空间上Sturm的距离，我们定义了可能不同总质量的公制度量空间之间的一类新的完整距离。 我们提供了几个明确的示例，这些例子是基于Hellinger-Kantorovich距离的地理指标扮演着重要角色。此外，我们讨论了该理论的一些限制案例，恢复了“纯运输” $ \ mathbf {d} $ - 距离，并引入了新的“纯熵”距离。 我们还详细研究了这种熵传输指标引起的拓扑，显示了满足RICCI曲率下限的度量度量空间的一些紧凑性和稳定性结果。
Inspired by the recent theory of Entropy-Transport problems and by the $\mathbf{D}$-distance of Sturm on normalised metric measure spaces, we define a new class of complete and separable distances between metric measure spaces of possibly different total mass. We provide several explicit examples of such distances, where a prominent role is played by a geodesic metric based on the Hellinger-Kantorovich distance. Moreover, we discuss some limiting cases of the theory, recovering the "pure transport" $\mathbf{D}$-distance and introducing a new class of "pure entropic" distances. We also study in detail the topology induced by such Entropy-Transport metrics, showing some compactness and stability results for metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense.
