Furstenberg's theorem for nonlinear stochastic systems
type: Item
instance of: scholarly article
title: Furstenberg's theorem for nonlinear stochastic systems
zbMATH Open document ID: 0592.60043
DOI: 10.1007/BF00363514
publication date: 1987-01-01T00:00:00Z
review text: We extend Furstenberg's theorem to the case of an i.i.d. random composition of incompressible diffeomorphisms of a compact manifold M. The original theorem applies to linear maps \(\{X_{i,i+1}\}_{i\in {\mathbb{N}}}\) on \({\mathbb{R}}^ m\) with determinant 1, and says that the highest Lyapunov exponent \[ \beta \equiv \lim_{n\to \infty}n^{-1}\| X_{n-1,n}\circ...\circ X_{0,1}\| \] is strictly positive unless there is a probability measure on the projective (m-1)-space which is a.s. invariant under the action of \(X_{i,i+1}\). Our extension refers to a probability measure on the projective bundle over M. We show that when our diffeomorphism is the flow of a stochastic differential equation, the criterion for \(\beta >0\) is ensured by a Lie algebra condition on the induced system on the principal bundle over M.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 93E15
zbMATH DE Number: 3951709
zbMATH Keywords: Furstenberg's theorem
zbMATH Keywords: random composition of incompressible diffeomorphisms
zbMATH Keywords: Lyapunov exponent
zbMATH Keywords: Lie algebra
zbMATH Keywords: principal bundle
author: Andrew Carverhill
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00363514
OpenAlex ID: W1985099948
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 3951709