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 minimizing movement schemes introduced by Jordan, Kinderlehrer and Otto. For locally Lipschitz continuous functionals which are $\lambda$-convex along generalized geodesics, we show that there exists a unique Wasserstein steepest descent flow which coincides with the Wasserstein gradient flow. 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：本文旨在实现两个目标。基于几何Wasserstein切空间，我们首先引入Wasserstein最速下降流。这些流是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。这允许使用欧拉向前格式，而非Jordan、Kinderlehrer和Otto引入的最小移动格式。对于在广义测地线上满足$\lambda$-凸性的局部Lipschitz连续泛函，我们证明存在唯一的Wasserstein最速下降流，且该流与Wasserstein梯度流一致。第二个目标是研究关于特定Riesz核的最大均值差异的Wasserstein流。关键部分在于相互作用能的处理。尽管该能量在广义测地线上不满足$\lambda$-凸性，我们仍给出了从Dirac测度出发的相互作用能Wasserstein最速下降流的解析表达式。与光滑核相反，粒子可能发生爆炸，即Dirac测度变为非Dirac测度。最速下降流的计算归结为寻找带外场的平衡测度，这巧妙地将相互作用能的Wasserstein流与位势理论联系起来。最后，我们提供了差异的Wasserstein最速下降流的数值模拟。