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Summary: AÆne Stochastic Di erential Equations with
In nite Delay on Abstract Phase Spaces
Markus Riedle 1
Institut fur Mathematik
Humboldt{Universitat zu Berlin
riedle@mathematik.hu-berlin.de
October 19, 2001
Abstract
A stochastic delay di erential equation is considered which is of the
form dX(t) = R
(1;0]
(ds)X(t + s)dt + dW (t), t 0, with the initial
condition X(u) = (u) for u 0. As it is successfully done in the deter-
ministic theory for delay equations, an axiomatic approach describing the
set of admissible initial functions is utilized to treat the stochastic equa-
tion, which permits the use of semi-group and spectral theory. Moreover,
it is obtained a representation of the solution in the abstract setting and
for a certain class of measures one can give suÆcient and necessary
conditions for the existence of a stationary solution.
1 Introduction
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