Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line

This paper provides results on Wasserstein gradient flows between measures on the real line. Utilizing the isometric embedding of the Wasserstein space ${\mathcal{P}}_2(\mathbb{R})$ into the Hilbert space $L_2((0,1))$, Wasserstein gradient flows of functionals on ${\mathcal{P}}_2(\mathbb{R})$ can be characterized as subgradient flows of associated functionals on $L_2((0,1))$. For the maximum mean discrepancy functional ${\mathcal{F}}_{\nu }:={\mathcal{D}}_K^2(\cdot ,\nu )$ with the non-smooth negative distance kernel $K(x,y)=-\mathrm{\mid }x-y\mathrm{\mid }$, we deduce a formula for the associated functional. This functional appears to be convex, and we show that ${\mathcal{F}}_{\nu }$ is convex along (generalized) geodesics. For the Dirac measure $\nu ={\delta }_q$, $q\in \mathbb{R}$ as end point of the flow, this enables us to determine the Wasserstein gradient flows analytically. Various examples of Wasserstein gradient flows are given for illustration.