The stable manifold theorem for non-linear stochastic systems with memory II. The local stable manifold theorem
نویسندگان
چکیده
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 infinitedimensional multiplicative ergodic theory techniques developed by D. Ruelle, together with interpolation arguments. r 2003 Elsevier Inc. All rights reserved. MSC: primary 60H10; 60H20; secondary 60H25
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