#### Abstract

We establish the existence of smooth densities for solutions of **R**^{d}-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, ∞) 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.

#### Recommended Citation

Bell, Denis R. and Mohammed, Salah-Eldin A. "Smooth Densities for Degenerate Stochastic Delay Equations with Hereditary Drift." (Jan 1995).

## Comments

Published in Annals of Probability, 23(4), 1875-1894.