A semigroup approach to stochastic delay equations in spaces of continuous functions

TY - JOUR

T1 - A semigroup approach to stochastic delay equations in spaces of continuous functions

AU - Riedle, Markus

AU - van Neerven, Jan

PY - 2007/4

Y1 - 2007/4

N2 - 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) 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.

AB - 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) 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.

KW - Stochastic integration in locally convex spaces

KW - stochastic delay equations in spaces of continuous functions

KW - invariant measures

U2 - 10.1007/s00233-005-0649-7

DO - 10.1007/s00233-005-0649-7

M3 - Article

SN - 0037-1912

VL - 72

SP - 227

EP - 239

JO - SEMIGROUP FORUM

JF - SEMIGROUP FORUM

IS - 2

ER -