On the exponential exit law in the small parameter exit problem

Abstract

We consider the diffusion dw in a domain D which contains a unique asymptotically stable critical point of the ODE . Using probabilistic estimates we prove the following: 1) The Principle eigenfunction of the differential generator for tghe process x(t converges to a constant as ∊→0, boundedly in D and uniformly on compacts. 2) If τ _D is the exit time of x(t) from D, then λτ _D converges in distribution to an exponential random variable with mean 1.(λ is the principle eigenvalue). Both of these results were known previosuly in the special case of a gradient flow: . Our arguments apply in the general non-gradient case.