Setvalued Dynamical Systems for Stochastic Evolution Equations Driven by Fractional Noise

We consider Hilbert-valued evolution equations driven by Hölder paths with Hölder index greater than 1/2, which includes the case of fractional noises with Hurst parameters in (1/2,1). The assumptions of the drift term will not be enough to ensure the uniqueness of solutions. Nevertheless, adopting a multivalued setting, we will prove that the set of all solutions corresponding to the same initial condition generates a (multivalued) nonautonomous dynamical system (Phi ). Finally, to prove that (Phi ) is measurable (and hence a (multivalued) random dynamical system), we need to construct a new metric dynamical system that models the noise with the property that the set space is separable.