Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion

In this paper, we consider a class of stochastic partial differential equations (SPDEs) driven by a fractional Brownian motion (fBm) with the Hurst parameter bigger than 1/2. The existence of local random unstable manifolds is shown if the linear parts of these SPDEs are hyperbolic. For this purpose we introduce a modified Lyapunov–Perron transform, which contains stochastic integrals. By the singularities inside these integrals we obtain a special Lyapunov–Perronʹs approach by treating a segment of the solution over time interval [0,1] as a starting point and setting up an infinite series equation involving these segments as time evolves. Using this approach, we establish the existence of local random unstable manifolds in a tempered neighborhood of an equilibrium.