I gave the following two talks at the Stochastic Analysis Seminar at the Mathematical Sciences Research Institute, Berkeley, California. The two talks describe recent joint work with Michael Scheutzow.
THE STABLE MANIFOLD THEOREM FOR SDE'S , Part I
Wednesday, December 3, 1997, 11:00-12:00 am, MSRI Lecture Hall.
In this talk, we formulate a local stable manifold theorem for stochastic differential equations in Euclidean space, driven by multi-dimensional Brownian motion. We introduce the concept of hyperbolicity for stationary trajectories of a SDE. This is done using the Oseledec muliplicative ergodic theorem on the linearized SDE around the stationary solution. Using methods of (non-linear ergodic theory), we construct a stationary family of stable and unstable manifolds in a stationary neighborhood around the hyperbolic stationary trajectory of the non-linear SDE. The stable/unstable manifolds are dynamically characterized using anticipating stochastic calculus.
SECOND TALK :
THE STABLE MANIFOLD THEOREM FOR SDE'S, Part II
Friday, December 5, 1997, 11:00-12:00 am, MSRI Lecture Hall.
This is a continuation of the talk given on Wednesday, December 3, 1997. We outline the basic ideas underlying the proof of the Stable Manifold Theorem for SDE's. We discuss the linearization of the SDE along a hyperbolic stationary solution. The stable and unstable manifolds are constructed using ideas and techniques from multiplicative ergodic theory that were developed by David Ruelle in the late seventies. In particular, we develop estimates of the stochastic flow in a neighborhood of the hyperbolic stationary solution. Finally, we discuss generalizations to semimartingale noise, related open problems, and conjectures.