Volterra equations perturbed by noise

We consider a linear abstract Volterra integrodifferential equation in a Hilbert space, forced by a Gaussian process. The equation involves a completely monotone convolution kernel with a singularity at t = 0 and a sectorial linear spatial operator. Existence and uniqueness of a weak solution is established. Furthermore we give conditions such that the solution converges to a stationary process. Our method consists in a state space setting so that the corresponding solution process is Markovian, and the tools of linear analytic semigroup theory can be utilized.