Stochastic flows on the boundaries of \(\text{SL}(n,\mathbb{R})\)
type: Item
instance of: scholarly article
title: Stochastic flows on the boundaries of \(\text{SL}(n,\mathbb{R})\)
zbMATH Open document ID: 0791.58110
DOI: 10.1007/BF01192136
author: Ming Liao
publication date: 1994-06-06T00:00:00Z
review text: We study the asymptotic stability of the stochastic flows on a class of compact spaces induced by a diffusion process in \(\text{SL}(n,\mathbb{R})\) or \(\text{GL}(n,\mathbb{R})\). These compact spaces are called boundaries of \(\text{SL}(n,\mathbb{R})\), which include \(\text{SO}(n)\), the flag manifold, the sphere \(S^{n-1}\) and the Grassmannians. The one point motions of these flows are Brownian motions. For almost every \(\omega\), we determine the set of stable points. This is a random open set whose complement has zero Lebesgue measure. The distance between any two points in the same component of this set tends to zero exponentially fast under the flow. The Lyapunov exponents at stable points are computed explicitly. We apply our results to a stochastic flow on \(S^{n-1}\) generated by a stochastic differential equation which exhibits some nice symmetry.
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 34D08
Mathematics Subject Classification ID: 37C75
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 569091
zbMATH Keywords: asymptotic stability
zbMATH Keywords: stochastic flows
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01192136
OpenAlex ID: W2080407173
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 569091