Convergence of alternating score-system iterations for regularized Landau particle dynamics

Abstract: We analyze an alternating iteration for score-based particle approximations of the homogeneous Landau equation. Each step computes a regularized score along the previous trajectory and recomputes the trajectory with that score frozen. First, at fixed positive regularization, we prove existence, uniqueness, and $W_1$-Lipschitz stability of a Tikhonov score minimizer in a vector-valued reproducing kernel Hilbert space embedded in $C_b<sup>2$. For bounded Lipschitz collision matrices, this yields factorial convergence to the coupled regularized dynamics on every finite interval, with constants independent of particle number. Global objective gaps and system residuals control inexact solves, and structural kernel assumptions ensure momentum and energy conservation. The mean-field and alternating-iteration limits commute at fixed regularization in this regime. Second, score-difference cancellation extends the theory to far-field-truncated Landau kernels with $γ\geq -2$ and the uncut case $γ=-2$. For the uncut three-dimensional Coulomb kernel, a separate continuum construction gives local factorial convergence from bounded, compactly supported densities. Third, we characterize the obstruction to removing regularization at fixed particle number and establish a separate conditional density-level convergence criterion under uniform regularity and approximation assumptions, using relative entropy. A heat-equation counterexample shows that an unregularized density-level Picard iteration can diverge. Atomic Coulomb and simultaneous particle-zero-regularization limits remain open.