  We study the long time statistics of a class of semi--linear damped wave
equations with polynomial nonlinearities and perturbed by additive Gaussian
noise in dimensions 2 and 3. We find that if sufficiently many directions in
the phase space are stochastically forced, the system is exponentially
attractive toward its unique invariant measure with a convergent rate that is
uniform with respect to the mass. Then, in the small mass limit, we prove the
convergence of the first marginal of the invariant measures in a suitable
Wasserstein distance toward the unique invariant measure of a stochastic
reaction--diffusion equation. This together with uniform geometric ergodcity
implies the validity of the small mass limit for the solutions on the infinite
time horizon $[0,\infty)$, thereby extending previously known results
established for the damped wave equations under Lipschitz nonlinearities.
