Wasserstein gradient flow of Maximum Mean Discrepancy with energy kernel

Wasserstein gradient flows of Maximum Mean Discrepancy have recently found applications in statistics and machine learning. The standard theory of Wasserstein gradient flows in spaces of probability measures provides well-posedness for smooth kernels, however, the kernels that perform well in applications are not differentiable and the theory does not apply. Here we consider the "energy kernels" $k(x,y) = -|x-y|^q$ for $q \in (0,2)$ which allow for long-range effects. We develop well-posedness theory in $L^p$ spaces and show that these solutions can be approximated well by interacting particles, which justifies their use. We also provide results that show that at least in 1D the convergence can be exponential when the solution is close to the target and also prove that no quantitative rate of convergence holds for general initial data. Joint work with Matthew Rosenzweig and Dejan Slepčev (CMU).