Wasserstein gradient flows for Coulomb discrepancies

Antonin Chodron de Courcel, Matthew Rosenzweig

We study the long-time behavior of the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) between a probability measure $ρ$ and a target measure $μ$, where the underlying kernel is given by a Coulomb potential. First, we establish the existence of global weak solutions starting from arbitrary Borel probability measures and prove an ultracontractive estimate, showing that the density $ρ_t$ becomes instantly bounded in $L^\infty$ for $t>0$. We also investigate the regularity of these solutions, showing that the Hölder norm can grow exponentially in time. Second, on the flat torus $\mathbb T^\mathsf{d}$, we prove exponential decay of the squared MMD along the flow toward a uniformly positive target $μ$, without requiring a lower bound on the initial data. This result is based on a ''defective Polyak-Lojasiewicz (PL) inequality'' whose defect term accounts for possible vacuum regions in the evolving density. We also prove that the usual PL inequality may fail when the target vanishes only at one point, and, in dimensions at least two, that no coercivity constant can depend only on a prescribed positive lower bound for the target. Finally, on $\mathbb R^\mathsf{d}$, we identify an obstruction at spatial infinity. For a compactly supported target, uniformly localized sources initially separated from the target by distance $D$ retain a fixed fraction of their initial squared MMD for times of order $D$. Consequently, neither a multiplicative squared-MMD decay modulus uniform over the initial datum nor a global PL inequality can hold on the unrestricted whole-space class. By contrast, under radial symmetry, source-support inclusion, and target-positivity assumptions, we establish a PL inequality and exponential convergence.