  We develop a systematic approach to the linear-noise approximation for
stochastic reaction systems with distributed delays. Unlike most existing work
our formalism does not rely on a master equation, instead it is based upon a
dynamical generating functional describing the probability measure over all
possible paths of the dynamics. We derive general expressions for the chemical
Langevin equation for a broad class of non-Markovian systems with distributed
delay. Exemplars of a model of gene regulation with delayed auto-inhibition and
a model of epidemic spread with delayed recovery provide evidence of the
applicability of our results.
