  This article deals with invariant manifolds for infinite dimensional random
dynamical systems with different time scales. Such a random system is generated
by a coupled system of fast-slow stochastic evolutionary equations. Under
suitable conditions, it is proved that an exponentially tracking random
invariant manifold exists, eliminating the fast motion for this coupled system.
It is further shown that if the scaling parameter tends to zero, the invariant
manifold tends to a slow manifold which captures long time dynamics. As
examples the results are applied to a few systems of coupled
parabolic-hyperbolic partial differential equations, coupled parabolic partial
differential-ordinary differential equations, and coupled hyperbolic-hyperbolic
partial differential equations.
