We consider the diffusiondw in a domainDwhich 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 processx(tconverges to a constant as ε→0, boundedly inDand uniformly on compacts. 2) If τDis the exit time ofx(t) fromD, then λτDconverges 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.