  The characteristic equation for a linear delay differential equation (DDE)
has countably infinite roots on the complex plane. This paper considers linear
DDEs that are on the verge of instability, i.e. a pair of roots of the
characteristic equation lie on the imaginary axis of the complex plane, and all
other roots have negative real parts. It is shown that, when small noise
perturbations are present, the probability law of the dynamics can be
approximated by the probability law of a one dimensional stochastic
differential equation (SDE) without delay. This is advantageous because
equations without delay are easier to simulate and one-dimensional SDE are
analytically tractable. When the perturbations are also linear, it is shown
that the stability depends on a specific complex number. The theory is applied
to study oscillators with delayed feedback. Some errors in other articles that
use multiscale approach are pointed out.
