Propagation of chaos for large Brownian particle system with Coulomb interaction

We investigate a system of N Brownian particles with the Coulomb interaction in any dimension d ≥ 2, and we assume that the initial data are independent and identically distributed with a common density ρ0 satisfying ∫ Rd ρ0 ln ρ0 dx < ∞ and ρ0 ∈ L 2d d+2 (Rd ) ∩ L1(Rd, (1+ |x|2) dx). We prove that there exists a unique global strong solution for this interacting partsicle system and there is no collision among particles almost surely. For d = 2, we rigorously prove the propagation of chaos for this particle system globally in time without any cutoff in the following sense. When N → ∞, the empirical measure of the particle system converges in law to a probability measure and this measure possesses a density which is the unique weak solution to the mean-field Poisson–Nernst–Planck equation of single component.