Invariant manifolds for stochastic partial dierential equations

Randomness or uncertainty is ubiquitous in scientic and engineering systems. Stochastic ef-fects are not just introduced to compensate for defects in deterministic models, but are often rather intrinsic phenomena. Taking stochastic eects into account is of central importance for the devel-opment of mathematical models of many phenomena in physics, mechanics, biology, economics and other disciplines. Macroscopic models in the form of partial dierential equations for these systems contain such randomness as stochastic forcing, uncertain parameters, random sources or inputs, and random initial and boundary conditions. Stochastic partial dierential equations are appropriate models for randomly in uenced systems. Although many useful techniques exist to investigate deterministic partial dierential equations as nonlinear dynamical systems, fundamental issues about studying stochastic partial dierential equations as random dynamical systems remain unsolved. In order to investigate stochastic partial dierential equations from a dynamical systems point of view, we need to establish a theory for invariant manifolds for stochastic partial dierential equations. As in deterministic systems, we expect that invariant manifolds, especially stable and unstable manifolds, to be essential for describing and understanding dynamical behavior of nonlinea