Wasserstein gradient flows for Coulomb discrepancies
This study investigates the long-time convergence behavior of the squared Maximum Mean Discrepancy (MMD) with a Coulomb kernel under Wasserstein gradient flows, with particular attention to challenges arising in vacuum regions and unbounded domains. By introducing a defective Polyak–Łojasiewicz inequality and leveraging the variational structure of MMD together with hypercontractivity estimates and Hölder regularity analysis, the authors establish the existence of global-in-time weak solutions emanating from arbitrary Borel probability measures and prove that these solutions instantaneously belong to \(L^\infty\) for any positive time. Key contributions include demonstrating exponential decay of MMD for positive target measures on the torus and uncovering fundamental obstructions to convergence in the whole space \(\mathbb{R}^d\), while establishing an exponential convergence mechanism under radial symmetry.