On the asymptotic relation between equilibrium density and exit measure in the exit problem

Abstract

We consider the diffusion dx(t) = b{x(t))dt-\-y/sa(x(t))dw in a domain D which is contained in the domain of attraction of an asymptotically stable critical point of x = b(x). Using a formula of Hasminskii for the equilibrium measure of x(-) we show that the asymptotic behaviors of the exit measure P[x?{i:D)edy] and the equilibrium density pE(y) are connected. The formula connecting the two is essentially the same as one derived by Matkowsky and Schuss using formal methods. The treatment here provides probabilistic insight into the Matkowsky-Schuss formula.