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Long-Time Behaviors of Mean-Field Interacting Particle Systems Related to McKean–Vlasov Equations

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Abstract

In this paper, we investigate concentration inequalities, exponential convergence in the Wasserstein metric \(W_{1}\), and uniform-in-time propagation of chaos for the mean-field weakly interacting particle system related to McKean–Vlasov equation. By means of the known approximate componentwise reflection coupling and with the help of some new cost function, we obtain explicit estimates for those three problems, avoiding the technical conditions in the known results. Our results apply to possibly multi-well confinement potentials, and interaction potentials W with bounded second mixed derivatives \(\nabla ^2_{xy}W\) which are not too big, so that there is no phase transition. Several examples are provided to illustrate the results.

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Acknowledgements

We are grateful to the referees of this paper. Their comments and suggestions improve much the readability and the organization of the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, People’s Republic of China

    Wei Liu

  2. Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, Hubei, People’s Republic of China

    Wei Liu

  3. Institute for Advanced Study in Mathematics, Harbin Institute of Technology, NO.92 West Da Zhi St., Harbin, People’s Republic of China

    Liming Wu & Chaoen Zhang

  4. Laboratoire de Mathématiques Blaise Pascal, CNRS-UMR 6620, Université Clermont-Auvergne (UCA), 63000, Clermont-Ferrand, France

    Liming Wu

Authors
  1. Wei Liu
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Correspondence to Wei Liu.

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Communicated by C.Mouhot.

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The first author is supported by NSFC (12071361, 11731009, 12131019), the Fundamental Research Funds for the Central Universities 2042020kf0217 and 2042020kf0031, and CSC.

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Liu, W., Wu, L. & Zhang, C. Long-Time Behaviors of Mean-Field Interacting Particle Systems Related to McKean–Vlasov Equations. Commun. Math. Phys. 387, 179–214 (2021). https://doi.org/10.1007/s00220-021-04198-5

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  • DOI: https://doi.org/10.1007/s00220-021-04198-5

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