The smoothness of laws of random flags and Oseledets spaces of linear stochastic differential equations
type: Item
instance of: scholarly article
title: The smoothness of laws of random flags and Oseledets spaces of linear stochastic differential equations
zbMATH Open document ID: 0924.60031
DOI: 10.1023/A:1008680717092
author: P. Imkeller
published in: Potential Analysis
publication date: 1999-11-07T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 1269758
zbMATH Keywords: Oseledets spaces
zbMATH Keywords: Oseledets flag
zbMATH Keywords: multiplicative ergodic theory
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: Hörmander theorem
reviewed by: Hans Crauel
MaRDI profile type: Publication
full work available at URL: a:1008680717092
OpenAlex ID: W1514049197
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description: scientific article; zbMATH DE number 1269758