Recent progress on the small parameter exit problem†

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Abstract
In this paper we will bring together some recent results of S.-J. Sheu, T.A. Darden and ourselves and develop their application to the small parameter exit problem of A.D. Wentzell and M.I. Freidlin. This problem concerns the asymptotic behavior of the exit distribution from a domain of attraction for an exponentially stable critical point of a dynamical system with an asymptotically small random perturbation. Recent results of Day and Darden on regularity properties of the so-called quasipotential function allow certain improvements and generalizations to be made in the work fo S.-J. Sheu on the asyptotic behavior of the equilibrium density. Applying these results to the exit problem through its connection with the equilibrium density, [2], we obtain a new theorem on the exit problem: Theorem 4 below. This theorem subsumes previous results and generalizes the conclusion of the Matkowksi-Schuss-Kamin approach from smooth to nonsmooth quasipotental functions. In all cases the exit problem is reduced to the asymptotic behavior of a Laplace integral. In particular there do exist examples in which the exit measures fail to converge as the small parameter vanishes