Propagation of chaos for large Brownian particle system with Coulomb interaction

Abstract

We investigate a system of N Brownian particles with the Coulomb interaction in any dimension $$d\ge 2$$ , and we assume that the initial data are independent and identically distributed with a common density $$\rho _0$$ satisfying $$\int _{\mathbb {R}^{d}}\rho _0\ln \rho _0\,\hbox {d}x<\infty $$ and $$\rho _0\in L^{\frac{2d}{d+2}} (\mathbb {R}^{d}) \cap L^1(\mathbb {R}^{d}, (1+|x|^2)\,\hbox {d}x)$$ . 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 $$N\rightarrow \infty $$ , 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.