We establish the existence of smooth densities for solutions of Rd-valued stochastic hereditary differential systems of the form
dx(t) = H(t,x)dt + g(t, x(t - r))dW(t).
In the above equation, W is an n-dimensional Wiener process, r is a positive time delay, H is a nonanticipating functional defined on the space of paths in Rd and g is an n x d matrix-valued function defined on [0, ∞) x Rd, such that gg* has degeneracies of polynomial order on a hypersurface in Rd. In the course of proving this result, we establish a very general criterion for the hypoellipticity of a class of degenerate parabolic second-order time-dependent differential operators with space-independent principal part.
Bell, Denis R. and Mohammed, Salah-Eldin A. "Smooth Densities for Degenerate Stochastic Delay Equations with Hereditary Drift." (Jan 1995).