Degenerate Stochastic Differential Equations, Flows and Hypoellipticity

In this article we study stochastic hereditary systems on Rd, their flows and regularity of their solutions with respect to d-dimensional Lebesgue measure. More specifically we will state and outline the proofs of several results on the following issues: (i)Existence of smooth densities for solutions of stochastic hereditary equations whose covariances degenerate polynomially (anywhere) on hypersurfaces in Rd. (ii)Existence of smooth densities for diffusions with degeneracies of infinite order on a collection of hypersurfaces in Rd. (iii)Extension and refinement of Hormander\u27s hypoellipticity theorem for a large class of highly degenerate second order parabolic operators: Hormander\u27s Lie algebra condition is allowed to fail exponentially fast on the degeneracy hypersurfaces, which are imbedded in submanifolds of dimension less than d. The exponential decay rate near the degeneracy surface is found to be optimal. Our proofs are based on the Malliavin calculus and require new sharp estimates for Ito processes in Euclidean space