Boundary Local Time and Small Parameter Exit Problems with Characteristic Boundaries

The exit problem for an asymptotically small random perturbation of a stable dynamical system $x(t)$ in a region D is considered. The connection between the distribution of the position of first exit and the equilibrium density of the perturbed system subject to reflection from the boundary of D is developed. Earlier work treated the case in which $x(t)$ enters D nontangentially. Here the case in which $x(t)$ is everywhere tangent to the boundary is examined. The “small-noise” asymptotics of the boundary local time turn out to be of primary importance.