Dynamics of Semilinear Stochastic Partial Differential Equations

The main objective of the talk is to characterize the pathwise local structure of solutions of semilinear stochastic partial differential equations (spde's) near stationary solutions. We first prove the existence of smooth compacting semiflows for semilinear stochastic partial differential equations. We then establish local stable manifold theorems for these infinite-dimensional stochastic dynamical systems. In particular, these results give a random family of Frechet smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary solution. The unstable and stable manifolds are stationary, asymptotically invariant under the stochastic semiflow and have fixed (non-random) finite dimension and codimension, respectively.The main objective of the talk is to characterize the pathwise local structure of solutions of semilinear stochastic partial differential equations (spde's) near stationary solutions. We first prove the existence of smooth compacting semiflows for semilinear stochastic partial differential equations. We then establish local stable manifold theorems for these infinite-dimensional stochastic dynamical systems. In particular, these results give a random family of Frechet smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary solution. The unstable and stable manifolds are stationary, asymptotically invariant under the stochastic semiflow and have fixed (non-random) finite dimension and codimension, respectively