INERTIAL MANIFOLDS AND STATIONARY MEASURES FOR STOCHASTICALLY PERTURBED DISSIPATIVE DYNAMICAL-SYSTEMS

We prove the existence of inertial manifolds for a semilinear dynamical system perturbed by additive 'white noise'. This manifold is generated by a certain predictable stationary vector process Phi(t)(omega). We study properties of this process as well as the properties of the induced finite-dimensional stochastic system on the manifold (inertial form). The results obtained allow us to prove for the original stochastic system a theorem on stabilization of stationary solutions to a unique invariant measure. This measure is uniquely defined by the probability distribution of the process Phi(t)(omega) and the form of the invariant measure corresponding to the inertial form.