Quantitative propagation of chaos in total variation for unregularized Vlasov--Riesz--Fokker--Planck systems

Source abstract

We establish quantitative propagation of chaos in total variation for the unregularized three-dimensional repulsive Vlasov--Poisson--Fokker--Planck (VPFP) particle system. For tensorized initial data satisfying weighted Sobolev regularity and a Gaussian moment, we prove that for every $0 0$ and $C_b\ge1$, independent of $N$ and $k$, such that $$\sup_{0\le t\leτ_b}\|F_{N,k}(t)-f_t^{\otimes k}\|_{\mathrm{TV}} \le C_b^kN^{-b}, \qquad N\ge1,\quad 1\le k\le N.$$ More generally, for repulsive Riesz potentials with singularity $|x|^{-s}$ on $\mathbb{T}^d$ and $\mathbb{R}^d$, $0 d/(d-s-1)$. The proof combines conditioning of the product initial law, auxiliary $k$-particle Fokker--Planck equations, and weighted $L^q$ estimates for the BBGKY hierarchy. The level-$(k+1)$ interaction term is controlled by a weighted Hölder estimate under the local condition $K\in L^{q'}_{\mathrm{loc}}$; for $d=3$ and $s=1$, this condition is $q>3$. On $\mathbb{R}^d$, polynomial spatial weights control the far field. For each fixed particle number, the singular particle dynamics are globally well posed and have no collisions. This permits the force regularization to be removed.