We formulate and outline a proof of the Local Stable Manifold Theorem for stochastic differential equations (SDE\u27s) 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
