The stable manifold theorem for stochastic differential equations

• 成果类型: Article
• 署名作者: Mohammed, SEA; Scheutzow, MKR
• 署名单位: Southern Illinois University System; Southern Illinois University; Technical University of Berlin
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1022677380
• 发表日期: 1999
• 页码: 615-652
• 关键词: multiplicative ergodic-theory; SEMIMARTINGALES; exponents; calculus; FLOWS; SPACE

摘要

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 Ito-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 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.