  The characteristic equation for a linear delay differential equation (DDE)
has countably infinite roots on the complex plane. We deal with linear DDEs
that are on the verge of instability, i.e. a pair of roots of the
characteristic equation (eigenvalues) lie on the imaginary axis of the complex
plane, and all other roots have negative real parts. We show that, when the
system is perturbed by small noise, under an appropriate change of time scale,
the law of the amplitude of projection onto the critical eigenspace is close to
the law of a certain one-dimensional stochastic differential equation (SDE)
without delay. Further, we show that the projection onto the stable eigenspace
is small. These results allow us to give an approximate description of the
delay-system using an SDE (without delay) of just one dimension. The proof is
based on the martingale problem technique.
