# MathSciDoc: An Archive for Mathematician ∫

#### Probabilitymathscidoc:1702.28002

Research in the Mathematical Sciences, 3, (1), 3-40, 2016.12
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.
Noncollision among particlesEntropy and Fisher information estimatesMartingale problemUniquenessde Finetti–Hewitt–Savage theorem
@inproceedings{jian-guo2016propagation,
title={Propagation of chaos for large Brownian particle system with Coulomb interaction},
author={Jian-Guo Liu, and Rong Yang},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170207210112150421262},
booktitle={Research in the Mathematical Sciences},
volume={3},
number={1},
pages={3-40},
year={2016},
}

Jian-Guo Liu, and Rong Yang. Propagation of chaos for large Brownian particle system with Coulomb interaction. 2016. Vol. 3. In Research in the Mathematical Sciences. pp.3-40. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170207210112150421262.