Stochastic Delay Equations with Hereditary Drift: Estimates of the Density

We consider a family of stochastic differential equations with a drift depending on the past history and a delayed diffusion term perturbed by a small parameter ε>0. We establish the asymptotic behaviour as ε↓0 for the logarithm of the corresponding family of densities at a fixed time t>0. The proof needs large deviation estimates and Malliavin calculus.