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

Authors:

Affiliation:

(1) Delft Institute of Applied Mathematics, Technical University of Delft, P.O. Box 5031, 2600 GA Delft, The Netherlands;(2) Department of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany

Abstract:

We present a semigroup approach to stochastic delay equations of the form

$dX(t)= left(int_{-h}^0X(t+s), dmu(s)right),dt + ,db(t)quadmbox{for }tge 0,$

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.

Keywords:

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