Open Access
Translator Disclaimer
April 1999 The Stable Manifold Theorem for Stochastic Differential Equations
Salah-Eldin A. Mohammed, Michael K. R. Scheutzow
Ann. Probab. 27(2): 615-652 (April 1999). DOI: 10.1214/aop/1022677380


We formulate and prove a local stable manifold theorem for stochastic differential equations (SDEs) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itôtype equations are treated. Starting with the existence of a stochastic flow for a SDE, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich SDEs, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating SDE. The proof of the stable manifold theorem is based on Ruelle–Oseledec multiplicative ergodic theory.


Download Citation

Salah-Eldin A. Mohammed. Michael K. R. Scheutzow. "The Stable Manifold Theorem for Stochastic Differential Equations." Ann. Probab. 27 (2) 615 - 652, April 1999.


Published: April 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0940.60084
MathSciNet: MR1698943
Digital Object Identifier: 10.1214/aop/1022677380

Primary: 60H10 , 60H20
Secondary: 60H05. , 60H25

Keywords: (perfect) cocycle , asymptotic invariance , hyperbolic stationary trajectory , local characteristics , local stable/unstable manifolds , Lyapunov exponents , spatial semimartingale , stochastic differential equation (SDE) , stochastic flow

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 2 • April 1999
Back to Top