  We establish a new criterion for exponential mixing of random dynamical
systems. Our criterion is applicable to a wide range of systems, including in
particular dispersive equations. Its verification is in nature related to
several topics, i.e., asymptotic compactness in dynamical systems, global
stability of evolution equations, and localized control problems.
  As an initial application, we exploit the exponential mixing of random
nonlinear wave equations with degenerate damping, critical nonlinearity, and
physically localized noise. The essential challenge lies in the fact that the
weak dissipation and randomness interact in the evolution.
