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In this paper, we explicitly calculate the quasi-potentials for the dampedsemilinear stochastic wave equation when the system is of gradient type. Weshow that in this case the infimum of the quasi-potential with respect to allpossible velocities does not depend on the density of the mass and doescoincide with the quasi-potential of the corresponding stochastic heat equationthat one obtains from the zero mass limit. This shows in particular that theSmoluchowski-Kramers approximation can be used to approximate long timebehavior in the zero noise limit, such as exit time and exit place from a basinof attraction.

Smoluchowski-Kramers approximation and large deviations for infinite dimensional gradient systems