A Semigroup Approach to Stochastic Delay Equations in Spaces of Continuous Functions

Abstract

We present a semigroup approach to stochastic delay equations of the form $$dX(t)= \left(\int_{-h}^0X(t+s)\, d\mu(s)\right)\,dt + \,d\b(t)\quad\mbox{for }t\ge 0,$$ $$X(t)= f(t)\quad\mbox{for } t\in [-h,0],$$ 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.