On smooth approximation for random attractor of stochastic partial differential equations with multiplicative noise

Wong–Zakai type approximation for stochastic partial differential equations (abbreviate as PDEs) is well studied. Besides the polygonal approximation, a type of smooth noise approximation is considered. After showing the existence of random attractor for a class of random partial differential equations defined on the entire space [Formula: see text], when random color noises tend to white noise, the solutions and invariant sets between original stochastic PDEs and random PDEs are compared. Some continuity results of random attractor in random dynamical systems are indicated.

Invariant Manifolds for Random and Stochastic Partial Differential Equations

Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic infuences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo- stable and pseudo-unstable manifolds for a class of random partial differential equations and stochastic partial differential equations is shown. Unlike the in- variant manifold theory for stochastic ordinary differential equations, random norms are not used. The result is then applied to a nonlinear stochastic partial differential equation with linear multiplicative noise.