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

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.