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 along 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 anticipative stochastic calculus
