The stable manifold theorem for stochastic differential equations
type: Item
instance of: scholarly article
title: The stable manifold theorem for stochastic differential equations
zbMATH Open document ID: 0940.60084
author: Salah Mohammed
author: Michael Scheutzow
published in: The Annals of Probability
publication date: 1999-11-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9803160
review text: Consider a stochastic differential equation on \(\mathbb{R}^d\) driven by a Kunita-type semimartingale with stationary ergodic increments, which may be either of Itô or of Stratonovich type. Suppose that the stochastic flow induced by the equation has a hyperbolic stationary (possibly non-adapted) trajectory, with hyperbolicity meaning that none of the associated Lyapunov exponents vanishes. Under these conditions existence of stable and unstable manifolds around the hyperbolic stationary trajectory is proved. The proof rests on arguments invoked in the approach of \textit{D. Ruelle} [Publ. Math., Inst. Hautes Étud. Sci. 50, 27-58 (1979; Zbl 0426.58014)], which are adapted for stochastic flows in continuous time. A key ingredient is the verification of integrability conditions for the derivative of the flow around the stationary solution.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 37H15
Mathematics Subject Classification ID: 37C40
zbMATH DE Number: 1359672
zbMATH Keywords: semimartingale-helix
zbMATH Keywords: multiplicative ergodic theory
zbMATH Keywords: invariant manifolds
reviewed by: Hans Crauel
MaRDI profile type: Publication
OpenAlex ID: W2163088626
arXiv ID: math/9803160
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DOI: 10.1214/AOP/1022677380
description: scientific article