The Stable Manifold Theorem for Stochastic Systems with Memory (Probability Seminar, Université Henri Poincaré Nancy 1)

Authors

Salah-Eldin A. Mohammed, Southern Illinois University CarbondaleFollow

Comments

Probability Seminar; Institut Élie Cartan; University Université Henri Poincaré Nancy 1; Nancy, France; June 17, 1999

Abstract

We state and prove a Local Stable Manifold Theorem for nonlinear stochastic differential systems with finite memory (viz. stochastic functional differential equations (sfde's)). We introduce the notion of hyperbolicity for stationary solutions of sfde's. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary solution. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses infinite-dimensional multiplicative ergodic theory techniques and interpolation arguments.