  We continue our study of the problem of mixing for a class of PDEs with very
degenerate noise. As we established earlier, the uniqueness of stationary
measure and its exponential stability in the dual-Lipschitz metric holds under
the hypothesis that the unperturbed equation has exactly one globally stable
equilibrium point. In this paper, we relax that condition, assuming only global
controllability to a given point. It is proved that the uniqueness of a
stationary measure and convergence to it are still valid, whereas the rate of
convergence is not necessarily exponential. The result is applicable to
randomly forced parabolic-type PDEs, provided that the deterministic part of
the external force is in general position, ensuring a regular structure for the
attractor of the unperturbed problem. The proof uses a new idea that reduces
the verification of a stability property to the investigation of a conditional
random walk.
