Stochastic delay equations with hereditary drift: Estimates of the density

We consider a family of stochastic differential equations with a drift depe nding on the past history and a delayed diffusion term perturbed by a small parameter epsilon > 0. We establish the asymptotic behaviour as epsilon do wn arrow 0 for the logarithm of the corresponding family of densities at a fixed time t > 0. The proof needs large deviation estimates and Malliavin c alculus. (C) 2000 Academic Press.