Target-adapted Green-Bessel SVGD: uniform-in-time propagation of chaos and last-iterate consistency

Abstract

We prove uniform-in-time propagation of chaos and last-iterate consistency for a target-adapted Stein variational gradient descent (SVGD) flow on compact connected manifolds. The target has a smooth positive density, and the particles start independently from a fixed smooth nonnegative density ratio. The construction uses the Green--Bessel operator $Q_{r,π}=A_π^{-1}(\mathrm{Id}+A_π)^{-r}$ of the reversible target Langevin generator. Sufficient Bessel smoothing gives a scalar kernel with finite diagonal, and a positive matrix lift realizes its potential force as a Stein velocity. Population and empirical flows then dissipate the same finite target discrepancy. Population entropy and the target spectral gap give decay of this discrepancy; a finite-time particle comparison reaches a time after which common-energy monotonicity controls every later time. The resulting expected uniform discrepancy is $O((\log N)^{-1/2})$, with a corresponding logarithmic $W_1$ bound and consistency along every sequence $t_N\to\infty$. We also prove an exact finite-mode approximation theorem with an explicit spatial-resolution error and a feature-factorized particle implementation. For confining Euclidean targets, we establish static kernel and moment results and give a conditional dynamical extension under explicit population-regularity and transport hypotheses.