Solutions of stochastic delay equations in spaces of continuous functions

Abstract

We present a semigroup approach to stochastic delay equations of the form dX(t) = (∫ 0 −h X(t+ s) dμ(s) ) dt+ dW (t) for t ≥ 0, X(t) = f(t) for t ∈ [−h, 0], in the space of continuous functions C[−h, 0]. We represent the solution as a C[−h, 0]valued process arising from a stochastic weak∗-integral in the bidual C[−h, 0]∗∗ and show how this process can be interpreted as a mild solution of an associated stochastic abstract Cauchy problem. We obtain a necessary and sufficient condition guaranteeing the existence of an invariant measure.