Small delay approximation of stochastic delay differential equations

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 univariate 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 also obtained using an average of the SDDE's drift term over the delayed dynamical variable, which defines a conditional average drift. This second approximation is characterized by the fact that the diffusion term is identical to that of the original SDDE. For small delays, our approach yields a steady-state probability density and a conditional average drift which are in close agreement with numerical simulations of the original SDDE. We illustrate this scheme with the delayed linear Langevin equation and a stochastic version of the delayed logistic equation. The technique can be used with any type of noise, and is easily generalized to multiple delays.