## The Annals of Probability

### Smooth Densities for Degenerate Stochastic Delay Equations with Hereditary Drift

#### Abstract

We establish the existence of smooth densities for solutions of $\mathbf{R}^d$-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 $\mathbf{R}^d$ and $g$ is an $n \times d$ matrix-valued function defined on $\lbrack 0,\infty) \times \mathbf{R}^d$, such that $gg^\ast$ has degeneracies of polynomial order on a hypersurface in $\mathbf{R}^d$. 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.

#### Article information

Source
Ann. Probab. Volume 23, Number 4 (1995), 1875-1894.

Dates
First available in Project Euclid: 19 April 2007

http://projecteuclid.org/euclid.aop/1176987807

Digital Object Identifier
doi:10.1214/aop/1176987807

Mathematical Reviews number (MathSciNet)
MR1379172

Zentralblatt MATH identifier
0852.60063

JSTOR