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We study the long time statistics of a class of semi--linear wave equationsmodeling the motions of a particle suspended in continuous media while beingsubjected to random perturbations via an additive Gaussian noise. By comparisonwith the nonlinear reaction settings, of which the solutions are known topossess geometric ergodicity, we find that, under the impact of nonlineardissipative damping, the mixing rate is at least polynomial of any order. Thisrelies on a combination of Lyapunov conditions, the contracting property of theMarkov transition semigroup as well as the notion of $d$--small sets.

Polynomial mixing of a stochastic wave equation with dissipative damping