Smooth densities for degenerate stochastic delay equations with hereditary drift

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 R(d) and g is an n X d matrix-valued function defined on [0, infinity) X R(d), such that gg* has degeneracies of polynomial order on a hypersurface in R(d). 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.