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

We discuss the long time statistics of a class of semi-linear damped wave equations with polynomial nonlinearities and random perturbations via an additive Gaussian noise. Under suitable conditions on the nonlinearities and stochastic forcing, we demonstrate that the system admits a unique invariant probability measure, which is exponentially attractive with a convergent rate independent of the mass. Then, in the small mass limit, we establish the convergence of the statistically steady states toward the unique invariant probability measure of a stochastic reaction-diffusion equation. As a consequence, we obtain the validity of the small mass limit for the solutions on the infinite time horizon.