Journal
JOURNAL OF FUNCTIONAL ANALYSIS
Volume 286, Issue 2, Pages -Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jfa.2023.110227
Keywords
Stable manifold; Interpolation space; Lyapunov-Perron method; Smoothness
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This article contributes to the understanding of stable manifolds for parabolic SPDEs driven by nonlinear multiplicative fractional noise. It proves the existence and smoothness of local stable manifolds through interpolation theory and the construction of a suitable function space.
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 man-ifolds for a class of parabolic SPDEs driven by nonlinear multiplicative fractional noise. We emphasize that stable man-ifolds 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 interpola-tion 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. (c) 2023 Elsevier Inc. All rights reserved.
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