  We investigate the convergence, in the small mass limit, of the stationary
solutions of a class of stochastic damped wave equations, where the friction
coefficient depends on the state and the noisy perturbation if of
multiplicative type. We show that the Smoluchowski-Kramers approximation that
has been previously shown to be true in any fixed time interval, is still valid
in the long time regime. Namely we prove that the first marginals of any
sequence of stationary solutions for the damped wave equation converge to the
unique invariant measure of the limiting stochastic quasilinear parabolic
equation. The convergence is proved with respect to the Wasserstein distance
associated with the $H^{-1}$ norm.
