Explicit asymptotic solutions for a class of weak-noise escape problems

This paper exploits the large deviations approach for estimating the first-exit time for a class of non-linear oscillatory systems with energy-dependent dissipation and weak noise. In sharp contrast to the great majority of large deviation problems, for oscillatory systems one can construct explicit solutions. We identify the Hamilton-Jacobi-Bellman equation whose solution characterizes the logarithmic asymptotics of the first-exit time and specify the types of the coefficients that allow the closed-form solutions. Examples of applications to engineering models illustrate the theoretical conclusions.