Large deviations results for the exit problem with characteristic boundary

Abstract

We consider the exit problem for an asymptotically small random perturbation of a stable dynamical system in a region D. We show the standard large deviations results for the exit distribution and mean exit time, as obtained by Wentzell and Freidlin under the assumption of nontangential drift 〈 b, n〉 < 0, can be generalized to the case of 〈 b, n〉 = 0 on the boundary. The method is to apply the approach of Wentzell and Freidlin in the context of a conormally reflected version of the small noise diffusion. Some simple examples are described which illustrate the point that large deviations results alone will not be adequate to resolve the asymptotic behavior of the exit distribution in many natural problems.