The smoothness of laws of random flags and Oseledets spaces of linear stochastic differential equations

The Oseledets spaces of a random dynamical system generated by a linear sto chastic differential equation are obtained as intersections of the correspo nding nested invariant spaces of a forward and a backward flag, described a s the stationary states of flows on corresponding flag manifolds. We study smoothness of their laws and conditional laws by applying Malliavin's calcu lus. 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-infinity-smooth. As an application we fin d that the semimartingale property is well preserved if the Wiener filtrati on is enlarged by the information present in the flag or Oseledets spaces.