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Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise

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2024

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Lin, Xiaofang
Zeng, Caibin

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National Natural Science Foundation of China: 11871225
National Natural Science Foundation of China: 12271177

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Journal of Functional Analysis. Elsevier. 2024, 286(2), 110227. ISSN 0022-1236. eISSN 1096-0783. Available under: doi: 10.1016/j.jfa.2023.110227

Zusammenfassung

Little seems to be known about the invariant manifolds for stochastic partial differential equations (SPDEs) driven by nonlinear multiplicative noise. Here we contribute to this aspect and analyze the Lu-Schmalfuß conjecture [Garrido-Atienza, et al., (2010) [14]] on the existence of stable manifolds for a class of parabolic SPDEs driven by nonlinear multiplicative fractional noise. We emphasize that stable manifolds for SPDEs are infinite-dimensional objects, and the classical Lyapunov-Perron method cannot be applied, since the Lyapunov-Perron operator does not give any information about the backward orbit. However, by means of interpolation theory, we construct a suitable function space in which the discretized Lyapunov-Perron-type operator has a unique fixed point. Based on this we further prove the existence and smoothness of local stable manifolds for such SPDEs.

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510 Mathematik

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ISO 690LIN, Xiaofang, Alexandra BLESSING-NEAMTU, Caibin ZENG, 2024. Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise. In: Journal of Functional Analysis. Elsevier. 2024, 286(2), 110227. ISSN 0022-1236. eISSN 1096-0783. Available under: doi: 10.1016/j.jfa.2023.110227
BibTex
@article{Lin2024Exist-68868,
  year={2024},
  doi={10.1016/j.jfa.2023.110227},
  title={Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise},
  number={2},
  volume={286},
  issn={0022-1236},
  journal={Journal of Functional Analysis},
  author={Lin, Xiaofang and Blessing-Neamtu, Alexandra and Zeng, Caibin},
  note={Article Number: 110227}
}
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