The stable manifold theorem for non-linear stochastic systems with memory: II. The local stable manifold theorem |
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Authors: | Salah-Eldin A. Mohammed Michael K.R. Scheutzow |
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Affiliation: | a Department of Mathematics, Southern Illinois University at Carbondale, Carbondale, IL 62901, USA b Fakultät II, Institut für Mathematik, MA 7-5, Technical University of Berlin, Strasse des 17 Juni 136, D-10623 Berlin, Germany |
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Abstract: | We state and prove a Local Stable Manifold Theorem (Theorem 4.1) for non-linear stochastic differential systems with finite memory (viz. stochastic functional differential equations (sfde's)). We introduce the notion of hyperbolicity for stationary trajectories of sfde's. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary trajectory. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle, together with interpolation arguments. |
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Keywords: | primary 60H10 60H20 secondary 60H25 |
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