On the Laws of the Oseledets Spaces of Linear Stochastic Differential Equations

Abstract

In many cases, the asymptotic properties of random dynamical systems generated by (non-linear) stochastic differential equations are determined by their invariant manifolds, and thus, to first order, by the linear invariant manifolds related to their linearizations. These were first described in the famous multiplicative ergodic theorem by Oseledets [Os] and are called “Oseledets spaces”. Although they are the direct analogue of eigenspaces for real matrices in the deterministic setting, they can explicitly be given only in rare cases (see Arnold [Ar]), and are hardly accessible. In this note we shall for simplicity from the very beginning consider linear stochastic differential equations instead of linearizations for non-linear ones, and describe some smoothness results for the laws of their Oseledets spaces, using the approach of Malliavin’s calculus.KeywordsLyapunov ExponentStochastic Differential EquationInvariant ManifoldGrassmannian ManifoldWiener SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.