Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D

TL;DR

This paper proves the propagation of molecular chaos for a 1D particle system with Coulomb interactions, establishing quantitative convergence rates and concentration inequalities towards the Vlasov-Poisson-Fokker-Planck equation.

Contribution

It introduces a trajectorial propagation of chaos result with optimal convergence rate and adapts weak-strong stability estimates for the particle system.

Findings

Convergence rate of order N^{-1/2} in expectation.

Exponential concentration inequalities for empirical measures.

Adaptation of stability estimates to the particle system.

Abstract

We consider a particle system in 1D, interacting via repulsive or attractive Coulomb forces. We prove the trajectorial propagation of molecular chaos towards a nonlinear SDE associated to the Vlasov-Poisson-Fokker-Planck equation. We obtain a quantitative estimate of convergence in expectation, with an optimal convergence rate of order $N^{-1/2}$. We also prove some exponential concentration inequalities of the associated empirical measures. A key argument is a weak-strong stability estimate on the (nonlinear) VPFP equation, that we are able to adapt for the particle system in some sense.