The smoothness of laws of random flags and Oseledets spaces of linear stochastic differential equations
Consider the linear SDE \(d\Phi- A_0\Phi dt+ \sum_{i=1}^m A_i\Phi\circ dW_i\) on \(\mathbb{R}^d\), where \(A_i\), \(0\leq i\leq m\), are \(d\times d\) matrices, and \(W_i\), \(1\leq i\leq m\), are (two-sided time) Wiener processes. With \(\lambda_1>\dots> \lambda_p\) denoting the Lyapunov exponents associated with this SDE, \(1\leq p\leq d\), consider the associated forward and backward flag spaces, resp. random linear subspaces. For instance, the \(j\)th forward flag space consists of all initial conditions in \(\mathbb{R}^d\) whose exponential growth rate for \(t\to\infty\) does not exceed \(\lambda_j\). The flag spaces can be interpreted as random variables in Grassmann manifolds of suitable dimensions, which are equipped with a Riemannian induced by a certain embedding into \(\mathbb{R}^{d\times d}\), the embedding being essentially obtained by interpreting a subspace as an orthogonal projection. It is first proved that the distributions as well as certain conditional distributions of the flag spaces are \(C^\infty\) smooth with respect to the Riemannian measure, provided the coefficient matrices \(A_i\) satisfy a nondegeneracy condition (which is satisfied, in particular, if the Lie algebra spanned by \(\{A_1,\dots, A_m\}\) is full). The arguments use Malliavin's calculus. Then the Oseledets splitting is investigated, which is obtained by taking intersections of forward and backward flag spaces, yielding random subspaces consisting of all initial conditions whose exponential growth rates coincide for \(t\to\infty\) and for \(t\to -\infty\) . For the Oseledets spaces smoothness of their (joint) distributions is derived from the corresponding results for appropriate conditional distributions of the flag spaces. The proof proceeds essentially by invoking arguments from geometric measure theory.
scientific article; zbMATH DE number 1269758