Random Dynamical Systems for Stochastic Evolution Equations Driven by Multiplicative Fractional Brownian Noise with Hurst Parameters H ∈ (1/3;1/2]

We consider the stochastic evolution equation du = Audt + G(u)dw, u(0) = u(0) in a separable Hilbert space V. Here G is supposed to be three times Frechet-differentiable and w is a trace class fractional Brownian motion with Hurst parameter H is an element of (1/3, 1/2]. We prove the existence of a unique pathwise global solution, and, since the considered stochastic integral does not produce exceptional sets, we are able to show that the above equation generates a random dynamical system.