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Pullback measure attractors for non-autonomous stochastic lattice systems

Part of: Infinite-dimensional dissipative dynamical systems Equations and systems with randomness Stochastic analysis

Published online by Cambridge University Press:  22 November 2024

Shaoyue Mi ,
Dingshi Li  and
Tianhao Zeng
Shaoyue Mi
Affiliation:
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, P. R. China (mishaoyue@my.swjtu.edu.cn (S. Mi)) (corresponding authors)
Dingshi Li
Affiliation:
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, P. R. China
Tianhao Zeng
Affiliation:
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, P. R. China
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Abstract

The aim of this article is to study the asymptotic behaviour of non-autonomous stochastic lattice systems. We first show the existence and uniqueness of a pullback measure attractor. Moreover, when deterministic external forcing terms are periodic in time, we show the pullback measure attractors are periodic. We then study the upper semicontinuity of pullback measure attractors as the noise intensity goes to zero. Pullback asymptotic compact for a family of probability measures with respect to probability distributions of the solutions is demonstrated by using uniform a priori estimates for far-field values of solutions.

Keywords

dynamical behaviournon-autonoumouspullback measure attractorstochastic latticeasymptotic compact

MSC classification

Primary: 37L55: Infinite-dimensional random dynamical systems; stochastic equations
Secondary: 34F05: Equations and systems with randomness 37L30: Attractors and their dimensions, Lyapunov exponents 60H10: Stochastic ordinary differential equations

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 20
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Bates, P. W., Lisei, H. and Lu, K.. Attractors for stochastic lattice dynamical systems. Stoch. Dyn. 6 (2006), 1–21.CrossRefGoogle Scholar
Bates, P. W., Lu, K. and Wang, B.. Attractors of non-autonomous stochastic lattice systems in weighted spaces. Phys. D Nonlinear Phenomena 289 (2014), 32–50.CrossRefGoogle Scholar
Caraballo, T., Han, X., Schmalfuss, B. and Valero, J.. Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise. Nonlinear Anal. 130 (2016), 255–278.CrossRefGoogle Scholar
Caraballo, T. and Lu, K.. Attractors for stochastic lattice dynamical systems with a multiplicative noise. Front. Math. China 3 (2008), 317–335.CrossRefGoogle Scholar
Caraballo, T., Morillas, F. and Valero, J.. Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities. J. Differ. Equ. 253 (2012), 667–693.CrossRefGoogle Scholar
Chen, Z., Li, X. and Wang, B.. Invariant measures of stochastic delay lattice systems. Discrete & Contin. Dyn. Syst. B 26 (2021), 3235–3269.Google Scholar
Crauel, H.. Measure attractors and Markov attractors. Dyn. Syst. 23 (2008), 75–107.CrossRefGoogle Scholar
Han, X., Kloeden, P. E. and Usman, B.. Long term behavior of a random Hopfield neural lattice model. Communications on Pure & Applied Analysis 18 (2019), 809–924.Google Scholar
Li, D. and Wang, B.. Pullback measure attractors for non-autonomous stochastic reaction-diffusion equations on thin domains. Preprint.Google Scholar
Li, D., Wang, B. and Wang, X.. Periodic measures of stochastic delay lattice systems. J. Differ. Equ. 272 (2021), 74–104.CrossRefGoogle Scholar
Li, D., Wang, B. and Wang, X.. Limiting behavior of invariant measures of stochastic delay lattice systems. J. Dyn. Differ. Equ. 34 (2022), 1453–1487.CrossRefGoogle Scholar
Marek, C. and Cutland, N. J.. Measure attractors for stochastic Navier-Stokes equations. Electron. J. Probab. 8 (1998), 1–15.Google Scholar
Morimoto, H.. Attractors of probability measures for semilinear stochastic evolution equations. Stoch. Anal. Appl. 10 (1992), 205–212.CrossRefGoogle Scholar
Schmalfuss, B.. Qualitative properties for the stochastic Navier-Stokes equation. Nonlinear Analysis: Theory, Methods & Applications 28 (1997), 1545–1563.CrossRefGoogle Scholar
Schmalfuß, B.. Long-time behaviour of the stochastic Navier-Stokes equation. Mathematische Nachrichten 152 (1991), 7–20.CrossRefGoogle Scholar
Schmalfuß, B.. Measure attractors and random attractors for stochastic partial differential equations. Stoch. Anal. Appl. 17 (1999), 1075–1101.CrossRefGoogle Scholar
Sui, M., Wang, Y., Han, X. and Kloeden, P. E.. Random recurrent neural networks with delays. J. Differ. Equ. 269 (2020), 8597–8639.CrossRefGoogle Scholar
Wang, B.. Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J. Differ. Equ. 253 (2012), 1544–1583.CrossRefGoogle Scholar
Wang, B.. Dynamics of stochastic reaction-diffusion lattice systems driven by nonlinear noise. J. Math. Anal. Appl. 477 (2019), 104–132.CrossRefGoogle Scholar
Wang, R. and Wang, B.. Random dynamics of p-Laplacian lattice systems driven by infinite-dimensional nonlinear noise. Stochastic Process. Appl. 130 (2020), 7431–7462.CrossRefGoogle Scholar
Wang, X., Lu, K. and Wang, B.. Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise. J. Dyn. Differ. Equ. 28 (2016), 1309–1335.CrossRefGoogle Scholar
Zhou, S.. Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise. J. Differ. Equ. 263 (2017), 2247–2279.CrossRefGoogle Scholar