#### Abstract

We formulate and outline a proof of the Local Stable Manifold Theorem for stochastic differential equations (SDE's) in Euclidean space (joint work with M. Scheutzow). This is a central result in dynamical systems with noise. Starting with the existence of a stochastic flow for an 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 a hyperbolic stationary solution. For Stratonovich noise, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating SDE.

*.dvi copy*

kanstalk2.ps (1203 kB)

*postscript copy*

kanstalk2.ps.Z (307 kB)

*compressed postscript copy*

## Comments

Ellis B. Stouffer Colloquium; Department of Mathematics; University of Kansas, October 1, 1998, Lawrence, Kansas