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 P2(mathbb R) into the Hilbert space L2((0,1)), Wasserstein gradient flows of functionals on mathcal P2(mathbb R) can be characterized as subgradient flows of associated functionals on L2((0,1)). For the maximum mean discrepancy functional mathcal F_nu := mathcal D2K(·, nu) with the non-smooth negative distance kernel K(x,y) = -|x-y|, 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 = δq, q ∈ 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.