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On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion

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Abstract

We study the generalized Dyson Brownian motion (GDBM) of an interacting N-particle system with logarithmic Coulomb interaction and general potential V. Under reasonable condition on V, we prove the existence and uniqueness of strong solution to SDE for GDBM. We then prove that the family of the empirical measures of GDBM is tight on \(\mathcal {C}([0,T],\mathscr {P}(\mathbb {R}))\) and all the large N limits satisfy a nonlinear McKean–Vlasov equation. Inspired by previous works due to Biane and Speicher, Carrillo, McCann and Villani, and Blower, we prove that the McKean–Vlasov equation is indeed the gradient flow of the Voiculescu free entropy on the Wasserstein space of probability measures over \(\mathbb {R}\). Using the optimal transportation theory, we prove that if \(V''\ge K\) for some constant \(K\in \mathbb {R}\), the McKean–Vlasov equation has a unique weak solution in the space of probability measures \(\mathscr {P}(\mathbb {R})\). This establishes the Law of Large Numbers and the propagation of chaos for the empirical measures of GDBM with non-quadratic external potentials which are not necessarily convex. Finally, we prove the longtime convergence of the McKean–Vlasov equation for \(C^2\)-convex potentials V.

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Notes

  1. In [34], Rogers and Shi proved the non-explosion of GDBM for V satisfying \(-xV'(x)\le \gamma \), \(\forall x\in \mathbb {R}\).

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Acknowledgements

The authors would like to express their sincere gratitude to the editor for his interest on this work, and to anonymous referees, who pointed out some technical problems in the earlier versions of this paper, and whose questions, comments and suggestions lead us to clarify and improve many places of this paper.

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Authors and Affiliations

  1. School of Mathematics, Renmin University of China, 59, Zhongguancun Da Jie, Beijing, 100872, China

    Songzi Li

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 55, Zhongguancun East Road, Beijing, 100190, China

    Xiang-Dong Li

  3. School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100049, China

    Xiang-Dong Li

  4. School of Mathematics and Statistics, Shandong Normal University, No. 1, University Road, Science and Technology Park, Changqing District, Ji’nan, 250385, China

    Yong-Xiao Xie

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  1. Songzi Li
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Correspondence to Xiang-Dong Li.

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Communicated by Eric A. Carlen.

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Songzi Li: Research supported by NSFC No. 11901569. Xiang-Dong Li: Research supported by NSFC No. 11771430, Key Laboratory RCSDS, CAS, No. 2008DP173182. Yong-Xiao Xie: Research supported by NSFC No. 11601287.

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Li, S., Li, XD. & Xie, YX. On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion. J Stat Phys 181, 1277–1305 (2020). https://doi.org/10.1007/s10955-020-02627-8

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  • DOI: https://doi.org/10.1007/s10955-020-02627-8

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