Propagation of chaos for large Brownian particle system with Coulomb interaction

Jian-Guo Liu Duke University Rong Yang Beijing University of Technology

Probability mathscidoc: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
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@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.
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