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Invariant Measures of Stochastic Lattice Plate Equations: Stability, Ergodicity and Mixing

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Abstract

This article is concerned with the stability, ergodicity and mixing of invariant measures of a class of stochastic lattice plate equations with nonlinear damping driven by a family of nonlinear white noise. The polynomial growth drift term has an arbitrary order growth rate, and the diffusion term is a family of locally Lipschitz continuous functions. By modifying and improving several energy estimates of the solutions uniformly for initial data when time is large enough, we prove that the noise intensity union of all invariant measures of the stochastic equation is tight on \(\ell ^2\times \ell ^2\). Then, we show that the weak limit of every sequence of invariant measures in this union must be an invariant measure of the corresponding limiting equation under the locally Lipschitz assumptions on the drift and diffusion terms. Under some globally Lipschitz conditions on the drift and diffusion terms, we also prove that every invariant measure of the stochastic equation must be ergodic and exponentially mixing in the pointwise and Wasserstein metric sense.

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Acknowledgements

Congli Yang was supported in part by CNSF (Grant No. 11861024). Renhai Wang was supported by National Natural Science Foundation of China (No. 12301299), the Guiyang City Science and Technology Plan Project (No.[2024]2-17), the research fund of Qianshixinmiao[2022]B16, Natural Science Research Project of Guizhou Provincial Department of Education (No. QJJ[2023]011), and the research fund of Qiankehepingtairencai-YSZ[2022]002. The authors would like to thank the reviewers for their valuable comments and suggestions on the paper.

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  1. School of Mathematics Science, Guizhou Normal University, Guiyang, 550025, China

    Yu Wang, Congli Yang & Renhai Wang

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  1. Yu Wang
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Correspondence to Renhai Wang.

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Communicated by Hongjun Gao.

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Wang, Y., Yang, C. & Wang, R. Invariant Measures of Stochastic Lattice Plate Equations: Stability, Ergodicity and Mixing. Bull. Malays. Math. Sci. Soc. 47, 87 (2024). https://doi.org/10.1007/s40840-024-01685-5

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  • DOI: https://doi.org/10.1007/s40840-024-01685-5

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