Smoluchowski-Kramers approximation for a general class of SPDEs

We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave equations mu u(tt) (t, x) +u(t) (t, x) = Delta u (t, x) +b(x, u (t, x)) +g (x, u (t, x)) (w) over dot(t), u (0) = u(o), u(t) (0) = v(o), endowed with Dirichlet boundary conditions, converges as the parameter mu goes to zero to the solution of the semi-linear stochastic heat equation u(t) (t, x) = Delta u (t, x) + b(x, u (t, x)) + g (x, u (t, x)) (w) over dot(t), u(0) = u(o), endowed with Dirichlet boundary conditions.