Regularity conditions for time-dependent MMD gradient flows

Investigate the specific regularity conditions on the time-dependent confinement potential V_t and interaction potential W_t in the free-energy formulation that are required to justify the derivation and well-posedness of maximum mean discrepancy (MMD) gradient flows in time-dependent reproducing kernel Hilbert spaces, beyond assuming existing conditions from Ferreira and Valencia-Guevara (2018).

Background

The paper extends MMD gradient flows to time-dependent RKHS by expressing the squared MMD as a free-energy functional with time-dependent potentials V_t and W_t. Using results from Ferreira and Valencia-Guevara (2018), the authors assume suitable regularity conditions to obtain a continuous probability path and an associated vector field via the continuity equation.

While this minimal derivation suffices for the experimental context, the authors explicitly defer a deeper analysis of the regularity assumptions. Clarifying these conditions would formalize when the time-dependent MMD flow exists and behaves well, strengthening the theoretical foundations of spectral mean flows.