  Dynamical systems driven by nonlinear delay SDEs with small noise can exhibit
important rare events on long timescales. When there is no delay, classical
large deviations theory quantifies rare events such as escapes from metastable
fixed points. Near such fixed points, one can approximate nonlinear delay SDEs
by linear delay SDEs. Here, we develop a fully explicit large deviations
framework for (necessarily Gaussian) processes $X_t$ driven by linear delay
SDEs with small diffusion coefficients. Our approach enables fast numerical
computation of the action functional controlling rare events for $X_t$ and of
the most likely paths transiting from $X_0 = p$ to $X_T=q$. Via linear noise
local approximations, we can then compute most likely routes of escape from
metastable states for nonlinear delay SDEs. We apply our methodology to the
detailed dynamics of a genetic regulatory circuit, namely the co-repressive
toggle switch, which may be described by a nonlinear chemical Langevin SDE with
delay.
