Pathwise solutions and attractors for retarded SPDEs with time smooth diffusion coefficients

In this paper we study the longtime dynamics of mild solutions to retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. As a preparation for this purpose we have to show the existence and uniqueness of a cocycle solution of such an equation. We do not assume that the noise is given in additive form or that it is a very simple multiplicative noise. However, we need some smoothing property for the coefficient in front of the noise. The main idea of this paper consists of expressing the stochastic integral in terms of non-stochastic integrals and the noisy path by using an integration by parts. This latter term causes that in a first moment only a local mild solution can be obtained, since in order to apply the Banach fixed point theorem it is crucial to have the H\"older norm of the noisy path to be sufficiently small. Later, by using appropriate stopping times, we shall derive the existence and uniqueness of a global mild solution. Furthermore, the asymptotic behavior is investigated by using the {\it Random Dynamical Systems theory}. In particular, we shall show that the global mild solution generates a random dynamical system that, under an appropriate smallness condition for the time lag, have associated a random attractor.