The small mass limit for long time statistics of a stochastic nonlinear damped wave equation

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.