A Jump-Modified Wasserstein Gradient Flow for Maximum Mean Discrepancy with Energy Kernels

Abstract

We study a jump-modified Wasserstein gradient flow of the squared maximum mean discrepancy generated by the energy kernels $K(z)=-|z|^q$, $0<q<2$, in every dimension $d\ge2$ as well as the one-dimensional range $1\le q<2$. Under subcritical density and finite-moment assumptions, we prove global well-posedness. The added jumps transfer mass toward smaller values of the first variation and dissipate the same MMD energy as the transport term. Under our hypotheses on the target and jump rates, every Borel probability measure with finite MMD energy and zero jump dissipation is equal to the target. Combining this rigidity with compactness and a second-moment identity yields convergence to the target from finite-second-moment data, without additional uniform-in-time density or moment assumptions. We also construct a global collision-free particle process whose energy is nonincreasing along each path. A quantitative mean-field estimate applies to general initial empirical approximations, including deterministic configurations; propagation of chaos for exchangeable data and long-time target approximation follow as consequences.