Random Dynamical Systems for Stochastic Evolution Equations Driven by Multiplicative Fractional Brownian Noise with Hurst Parameters $H{\in} (1/3,1/2

Abstract

We consider the stochastic evolution equation $du=Audt+G(u)d\omega,\quad u(0)=u_0$ in a separable Hilbert space $V$. Here $G$ is supposed to be three times Frechet-differentiable and $\omega$ is a trace class fractional Brownian motion with Hurst parameter $H\in (1/3,1/2]$. 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.