Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels

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., the 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. Furthermore, we prove convergence of the minimizing movement scheme to our Wasserstein steepest descent flow. Finally, we provide analytic Wasserstein steepest descent flows of discrepancies in one dimension and numerical simulations in two and three dimensions.