  In this paper, we study the quasi-potential for a general class of damped
semilinear stochastic wave equations. We show that, as the density of the mass
converges to zero, the infimum of the quasi-potential with respect to all
possible velocities converges to the quasi-potential of the corresponding
stochastic heat equation, that one obtains from the zero mass limit. This shows
in particular that the Smoluchowski-Kramers approximation is not only valid for
small time, but, in the zero noise limit regime, can be used to approximate
long-time behaviors such as exit time and exit place from a basin of
attraction.
