Delay differential equations evolve in an infinite-dimensional phase space.
In this paper, we consider the effect of external fluctuations (noise) on
delay differential equations involving one variable, thus leading to univar
iate stochastic delay differential equations (SDDE's). For small delays, a
univariate nondelayed stochastic differential equation approximating such a
SDDE is presented. Another approximation, complementary to the first, is a
lso obtained using an average of the SDDE's drift term over the delayed dyn
amical variable, which defines a conditional average drift. This second app
roximation is characterized by the fact that the diffusion term is identica
l to that of the original SDDE. For small delays, our approach yields a ste
ady-state probability density and a conditional average drift which are in
close agreement with numerical simulations of the original SDDE. We illustr
ate this scheme with the delayed linear Langevin equation and a stochastic
version of the delayed logistic equation. The technique can be used with an
y type of noise, and is easily generalized to multiple delays.