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Mean Attractors and Invariant Measures for Fractional Stochastic Lattice Systems Driven by Nonlinear Noise

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  • Pengyu Chen

    (Northwest Normal University
    Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics)

  • Ailin Bai

    (Northwest Normal University)

  • Xuping Zhang

    (Northwest Normal University)

Abstract

This paper is concerned with the existence of mean random attractors and invariant measures for fractional stochastic lattice systems driven by nonlinear noise. We firstly establish the global existence and uniqueness of solutions, and then prove the existence and uniqueness of weak pullback mean random attractors of the fractional stochastic lattice systems in the Bochner space $$L^{2}(\Omega ,\ell ^{2})$$ L 2 ( Ω , ℓ 2 ) . Under certain conditions, we establish the tightness of a family of distributions of solutions by using the uniform estimates on the tails of solutions and then show the existence of invariant measures of the system. We also discuss the limiting behavior of invariant measures of fractional stochastic lattice systems driven by nonlinear noise as $$\epsilon \rightarrow 0$$ ϵ → 0 . Finally, under further assumptions on the nonlinear terms, we show that the system has a unique, ergodic, mixing, and stable invariant probability measure in $$\ell ^{2}$$ ℓ 2 .

Suggested Citation

  • Pengyu Chen & Ailin Bai & Xuping Zhang, 2025. "Mean Attractors and Invariant Measures for Fractional Stochastic Lattice Systems Driven by Nonlinear Noise," Journal of Theoretical Probability, Springer, vol. 38(4), pages 1-50, December.
    • Handle: RePEc:spr:jotpro:v:38:y:2025:i:4:d:10.1007_s10959-025-01442-9
    DOI: 10.1007/s10959-025-01442-9
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    References listed on IDEAS

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    1. Manil T. Mohan & K. Sakthivel & Sivaguru S. Sritharan, 2019. "Ergodicity for the 3D stochastic Navier–Stokes equations perturbed by Lévy noise," Mathematische Nachrichten, Wiley Blackwell, vol. 292(5), pages 1056-1088, May.
    2. Wang, Renhai & Wang, Bixiang, 2020. "Random dynamics of p-Laplacian lattice systems driven by infinite-dimensional nonlinear noise," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7431-7462.
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