Publikation: Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise
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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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LIN, 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.2023.110227BibTex
@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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