论文标题
Geodesics和Geodesic $λ$ -Convexity for Hellinger-Kantorovich距离的良好特性
Fine properties of geodesics and geodesic $λ$-convexity for the Hellinger-Kantorovich distance
论文作者
论文摘要
我们研究了Hellinger双重配方的最佳势能的良好规律性 - Kantorovich问题（HK），为原始Monge配方的可溶性提供了足够的条件。我们还建立了新的规律性特性，用于在HK的双重动态公式中产生的汉密尔顿 - 雅各比方程的溶液，它们足够强，可以构建一种特征性的传输流动流，这些传输流动驱动了两种任意阳性度量之间的地质插值。这些结果应用于研究HK测量学的相关几何特性，并沿运输流程得出其Lebesgue密度的凸行为。最后，得出了在措施空间上定义的功能的确切条件，以保证相对于Hellinger-Kantorovich距离，地球$λ$ -Convexity。提供了大地凸功能的示例。
We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger--Kantorovich problem (HK), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton--Jacobi equation arising in the dual dynamic formulation of HK, which are sufficiently strong to construct a characteristic transport-dilation flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of HK geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic $λ$-convexity with respect to the Hellinger--Kantorovich distance. Examples of geodesically convex functionals are provided.
