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Averaging principle and normal deviations for multi-scale stochastic hyperbolic–parabolic equations

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Abstract

We study the asymptotic behavior of stochastic hyperbolic–parabolic equations with slow–fast time scales. Both the strong and weak convergence in the averaging principle are established. Then we study the stochastic fluctuations of the original system around its averaged equation. We show that the normalized difference converges weakly to the solution of a linear stochastic wave equation. An extra diffusion term appears in the limit which is given explicitly in terms of the solution of a Poisson equation. Furthermore, sharp rates for the above convergence are obtained, which are shown not to depend on the regularity of the coefficients in the equation for the fast variable.

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Acknowledgements

The authors would like to thank the referees for a careful reading of the first submitted version and for a number of useful comments and suggestions for improving the paper.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, 33501, Bielefeld, Germany

    Michael Röckner

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences (CAS), Beijing, 100190, People’s Republic of China

    Michael Röckner

  3. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221000, Jiangsu, People’s Republic of China

    Longjie Xie

  4. School of Mathematics, Shandong University, Jinan, 250100, Shandong, People’s Republic of China

    Li Yang

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  1. Michael Röckner
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  2. Longjie Xie
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  3. Li Yang
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Correspondence to Li Yang.

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This work is supported by the DFG through CRC 1283 and NSFC (Nos. 12071186, 11931004, 12090011).

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Röckner, M., Xie, L. & Yang, L. Averaging principle and normal deviations for multi-scale stochastic hyperbolic–parabolic equations. Stoch PDE: Anal Comp 11, 869–907 (2023). https://doi.org/10.1007/s40072-022-00248-8

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