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On Averaging and Mixing for Stochastic PDEs

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Abstract

We examine the convergence in the Krylov–Bogolyubov averaging for nonlinear stochastic perturbations of linear PDEs with pure imaginary spectrum and show that if the involved effective equation is mixing, then the convergence is uniform in time.

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Notes

  1. So \(\{\beta _l(t)\}\) are standard independent complex Brownian motions.

  2. The matrix \((A_{kl})\) defines a non-negative compact self-adjoint operator in the space \(l^2\). So its principal square root (which defines another non-negative compact self-adjoint operator) exists.

  3. In [2, Theorem 2.9] the uniform in time convergence as above is proved for a class of systems 2.14. The proof in [2] is based on the observation that the corresponding v-equations are mixing with a rate of mixing, independent from \(\varepsilon \).

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Acknowledgements

We are thankful to Armen Shirikyan for discussion.

Funding

GH was supported by National Natural Science Foundation of China (Significant project No.11790273). SK was supported by the Ministry of Science and Higher Education of the Russian Federation (megagrant No. 075-15-2022-1115).

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Authors and Affiliations

  1. Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

    Guan Huang

  2. Institut de Mathémathiques de Jussieu–Paris Rive Gauche, CNRS, UMR 7586, Sorbonne Paris Cité, Université Paris Diderot, 75013, Paris, France

    Sergei Kuksin

  3. Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, Russian Federation, 117198

    Sergei Kuksin

Authors
  1. Guan Huang
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  2. Sergei Kuksin
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Both authors wrote and reviewed the main text of the manuscript.

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Correspondence to Guan Huang.

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Huang, G., Kuksin, S. On Averaging and Mixing for Stochastic PDEs. J Dyn Diff Equat (2022). https://doi.org/10.1007/s10884-022-10202-w

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  • DOI: https://doi.org/10.1007/s10884-022-10202-w

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