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.
"Smoluchowski–Kramers approximation and large deviations for infinite-dimensional nongradient systems with applications to the exit problem." Ann. Probab. 44 (4) 2591 - 2642, July 2016. https://doi.org/10.1214/15-AOP1029