The Stable Manifold Theorem for Stochastic Differential Equations

We formulate and prove a {\it Local Stable Manifold Theorem\/} for stochastic differential equations (sde's) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itô-type equations are treated. Starting with the existence of a stochast...

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Bibliographic Details
Published inarXiv.org
Main Authors Mohammed, Salah-Eldin A, Scheutzow, Michael K R
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 18.03.1998
Subjects
Differential equations
Ergodic processes
Manifolds
Mathematical analysis
Theorems
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Summary:We formulate and prove a {\it Local Stable Manifold Theorem\/} for stochastic differential equations (sde's) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itô-type 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 sde's, 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.
ISSN:2331-8422