A Direct Approach to the Exit Problem

Abstract

This paper considers the problem of exit for a dynamical system driven by small white noise, from the domain of attraction of a stable state. A direct singular perturbation analysis of the forward equation is presented, based on Kramers’ approach, in which the solution to the stationary Fokker–Planck equation is constructed, for a process with absorption at the boundary and a source at the attractor. In this formulation the boundary and matching conditions fully determine the uniform expansion of the solution, without resorting to “external” selection criteria for the expansion coefficients, such as variational principles or the Lagrange identity, as in our previous theory. The exit density and the mean first passage time to the boundary are calculated from the solution of the stationary Fokker–Planck equation as the probability current density and as the inverse of the total flux on the boundary, respectively. As an application, a uniform expansion is constructed for the escape rate in Kramers’ problem of activated escape from a potential well for the full range of the dissipation parameter.