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.