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

Maxime Hauray (1) , Samir Salem (1)

1 I2M - Institut de Mathématiques de Marseille

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.

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Maxime Hauray, Samir Salem. Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D. 2015. ⟨hal-01257016⟩