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Smoothness of invariant manifolds and foliations for infinite dimensional random dynamical systems

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Abstract

In this paper, we investigate the smoothness of invariant manifolds and foliations for random dynamical systems with nonuniform pseudo-hyperbolicity in Hilbert spaces. We discuss on the effect of temperedness and the spectral gaps in the nonuniform pseudo-hyperbolicity so as to prove the existence of invariant manifolds and invariant foliations, which preserve the CN,τ(ω) HÖlder smoothness of the random system in the space variable and the measurability of the random system in the sample point. Moreover, we also prove that the stable foliation is CN−1,τ(ω) in the base point.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11501549, 11331007, 11971330, 11771307, 11831012 and 11726623) and the Fundamental Research Funds for the Central Universities (Grant No. YJ201646). The authors are very grateful to the anonymous referees for their careful reading and helpful suggestions, which have notably improved the paper.

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Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, 610064, China

    Jun Shen & Weinian Zhang

  2. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Kening Lu

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  1. Jun Shen
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  2. Kening Lu
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  3. Weinian Zhang
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Correspondence to Kening Lu.

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Shen, J., Lu, K. & Zhang, W. Smoothness of invariant manifolds and foliations for infinite dimensional random dynamical systems. Sci. China Math. 63, 1877–1912 (2020). https://doi.org/10.1007/s11425-019-1664-3

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