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Stochastic dynamics of a neural field lattice model with state dependent nonlinear noise

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Abstract

The well-posedness and long term dynamics of a stochastic non-autonomous neural field lattice system on vector-valued indices \({\mathbb {Z}}^d\) driven by state dependent nonlinear noise are investigated in a weighted space of infinite sequences. First the existence and uniqueness of a mean square solution to the lattice system is established under the assumptions that the nonlinear drift and diffusion terms are component-wise continuously differentiable with weighted equi-locally bounded derivatives. Then the existence and uniqueness of a tempered weak pullback mean random attractor associated with the solution is proved. Finally the existence of invariant measures for the neural field lattice system is obtained by uniform tail-estimates and Krylov–Bogolyubov’s method.

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Acknowledgements

This work is done when Xiaoli Wang visit the Department of Mathematics and Statistics at Auburn University. She would like to express her thanks to all people there for their kind hospitality.

Funding

This work is partially supported by FEDER-Junta de Andalucia project P18-FR-4509.

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Authors and Affiliations

  1. College of Science, Henan University of Technology, Zhengzhou, 450001, China

    Xiaoli Wang

  2. School of Mathematics and Statistics and Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Xiaoli Wang

  3. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36832, USA

    Peter E. Kloeden & Xiaoying Han

Authors
  1. Xiaoli Wang
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  2. Peter E. Kloeden
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  3. Xiaoying Han
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Correspondence to Xiaoying Han.

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Wang, X., Kloeden, P.E. & Han, X. Stochastic dynamics of a neural field lattice model with state dependent nonlinear noise. Nonlinear Differ. Equ. Appl. 28, 43 (2021). https://doi.org/10.1007/s00030-021-00705-8

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