The Smoothness of Laws of Random Flags and Oseledets Spaces of Linear Stochastic Differential Equations

Abstract

The Oseledets spaces of a random dynamical system generated by a linear stochastic differential equation are obtained as intersections of the corresponding nested invariant spaces of a forward and a backward flag, described as the stationary states of flows on corresponding flag manifolds. We study smoothness of their laws and conditional laws by applying Malliavin's calculus. If the Lie algebras induced by the actions of the matrices generating the system on the manifolds span the tangent spaces at any point, laws and conditional laws are seen to be C∞-smooth. As an application we find that the semimartingale property is well preserved if the Wiener filtration is enlarged by the information present in the flag or Oseledets spaces.