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Stationarity and uniform in time convergence for the graphon particle system

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  • Bayraktar, Erhan
  • Wu, Ruoyu

Abstract

We consider the long time behavior of heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Under suitable assumptions, including a certain convexity condition, we show the exponential ergodicity for both systems, establish the uniform-in-time law of large numbers for marginal distributions as the number of particles increases, and introduce the uniform-in-time Euler approximation. The precise rate of convergence of the Euler approximation is provided.

Suggested Citation

  • Bayraktar, Erhan & Wu, Ruoyu, 2022. "Stationarity and uniform in time convergence for the graphon particle system," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 532-568.
    • Handle: RePEc:eee:spapps:v:150:y:2022:i:c:p:532-568
    DOI: 10.1016/j.spa.2022.04.006
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    References listed on IDEAS

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    1. Bayraktar, Erhan & Wu, Ruoyu, 2021. "Mean field interaction on random graphs with dynamically changing multi-color edges," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 197-244.
    2. Gaoyue Guo & Jan Obloj, 2017. "Computational Methods for Martingale Optimal Transport problems," Papers 1710.07911, arXiv.org, revised Apr 2019.
    3. Bhamidi, Shankar & Budhiraja, Amarjit & Wu, Ruoyu, 2019. "Weakly interacting particle systems on inhomogeneous random graphs," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 2174-2206.
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    Cited by:

    1. Hamed Amini & Zhongyuan Cao & Agnès Sulem, 2025. "Graphon mean-field backward stochastic differential equations with jumps and associated dynamic risk measures," Finance and Stochastics, Springer, vol. 29(4), pages 1139-1194, October.
    2. Coppini, Fabio & De Crescenzo, Anna & Pham, Huyên, 2025. "Nonlinear Graphon mean-field systems," Stochastic Processes and their Applications, Elsevier, vol. 190(C).
    3. Bayraktar, Erhan & Wu, Ruoyu, 2023. "Graphon particle system: Uniform-in-time concentration bounds," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 196-225.
    4. Guillermo Alonso Alvarez & Erhan Bayraktar & Ibrahim Ekren, 2025. "Contracting a crowd of heterogeneous agents," Papers 2507.09415, arXiv.org, revised May 2026.

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