Propagation of chaos for the Vlasov–Poisson–Fokker–Planck equation with a polynomial cut-off

Abstract

We consider a N -particle system interacting through the Newtonian potential with a polynomial cut-off in the presence of noise in velocity. We rigorously prove the propagation of chaos for this interacting stochastic particle system. Taking the cut-off like N −δ with δ < 1/d in the force, we provide a quantitative error estimate between the empirical measure associated to that N -particle system and the solutions of the d-dimensional Vlasov-Poisson-Fokker-Planck system. We also study the propagation of chaos for the Vlasov-Fokker-Planck equation with less singular interaction forces than the Newtonian one.

Fractional Keller–Segel equation: Global well-posedness and finite time blow-up

“…and let ρ be a smooth solution to the (FKS) equation, Φ be a non-decreasing convex mapping, Ψ and ψ be defined respectively by (13) and (14) and b ∈ (1, p a ).…”

“…The point (i) of this Lemma has been extensively used in the literature (See for instance [12,11,18,30]). So has the point (ii) in the Newtonian case a = d and thus p a = ∞ (see for instance [29,19,13]). However to the best of the authors' knowledge, its generalization to a general Riesz interaction kernel a ∈ (0, d) is a novelty.…”

On the Mean-Field Limit for the Vlasov–Poisson–Fokker–Planck System

“…Where σ > 0, the random particle method for approximating the VPFP system with the Coulombian kernel was studied in [26], where the initial data was chose on a mesh and the cut-off parameter could be N −δ (0 < δ < 1 d ). Most recently, Carrillo et al [12] also investigated the singular VPFP system but with the i.i.d. initial data, and obtained the propagation of chaos through a cut-off of N −δ (0 < δ < 1 d ), which was a generalization of [39].…”

“…In this manuscript, we set σ > 0. Compared with the recent work [12], the main technical innovation of this paper is that we fully use the randomness coming from the initial conditions and the Brownian motions to significantly improve the cut-off. Note that in [12] the size of cut-off can be very close to but larger than N − 1 d .…”

“…Compared with the recent work [12], the main technical innovation of this paper is that we fully use the randomness coming from the initial conditions and the Brownian motions to significantly improve the cut-off. Note that in [12] the size of cut-off can be very close to but larger than N − 1 d . However we manage to reduce the cut-off size to be smaller than N − 1 d (see Remark 1.4), which is a sort of average minimal distance between N particles in dimension d. This manuscript significantly improves the ideas presented in [10].…”