The aim of this paper is twofold. Based on the geometric Wasserstein tangent space, we first introduce Wasserstein steepest descent flows. These are locally absolutely continuous curves in the Wasserstein space whose tangent vectors point into a steepest descent direction of a given functional. This allows the use of Euler forward schemes instead of Jordan--Kinderlehrer--Otto schemes. For $\lambda$-convex functionals, we show that Wasserstein steepest descent flows are an equivalent characterization of Wasserstein gradient flows. The second aim is to study Wasserstein flows of the maximum mean discrepancy with respect to certain Riesz kernels. The crucial part is hereby the treatment of the interaction energy. Although it is not $\lambda$-convex along generalized geodesics, we give analytic expressions for Wasserstein steepest descent flows of the interaction energy starting at Dirac measures. In contrast to smooth kernels, the particle may explode, i.e., a Dirac measure becomes a non-Dirac one. The computation of steepest descent flows amounts to finding equilibrium measures with external fields, which nicely links Wasserstein flows of interaction energies with potential theory. Finally, we provide numerical simulations of Wasserstein steepest descent flows of discrepancies.

翻译：摘要：本文旨在实现两个目标。首先，基于几何Wasserstein切空间，我们引入Wasserstein最速下降流。这些是Wasserstein空间中局部绝对连续的曲线，其切向量指向给定泛函的最速下降方向。这使得我们能够采用欧拉前向格式替代Jordan–Kinderlehrer–Otto格式。对于λ-凸泛函，我们证明Wasserstein最速下降流是Wasserstein梯度流的等价刻画。第二个目标是研究关于特定Riesz核的最大均值差异Wasserstein流。其中关键部分在于相互作用能的处理。尽管该能量沿广义测地线不满足λ-凸性，我们仍给出了从狄拉克测度出发的相互作用能Wasserstein最速下降流的解析表达式。与光滑核不同，粒子可能发生“爆炸”，即狄拉克测度变为非狄拉克测度。最速下降流的计算归结为寻找具有外场的平衡测度，这巧妙地将相互作用能的Wasserstein流与势理论联系起来。最后，我们提供了差异度量Wasserstein最速下降流的数值模拟结果。