Large Deviations Estimates of Escape Time for Lagrangian Systems

This paper is concerned with analysis of the asymptotic behavior of a Lagranian system with small noise effects. The domain of the system operation is supposed to be within the domain of attraction of an asymptotically stable point of the unperturbed system. If noise is weak, escape from the reference domain is a rare event associated with large deviations in the system. This paper uses an extension of large deviations theory to the degenerate systems to develop the escape time asymptotics for a weakly perturbed Lagrangian system. Estimation of the statistical quantities is reduced to minimization of an associated action functional. It is shown that, in the case of the Lagrangian system, the solution of the associated variational problem can be found in a closed form, as a function of the system and noise parameters. As an example, motion of a 2n-dimensional linear system in an ellipsoidal domain is studied. Application of the theory to the nonlinear systems is illustrated by estimation of the lifetime of the Henon-Heiles system.