On the exit law from saddle points

We consider the effects of adding an asymptotically small random (Brownian) perturbation to a planar dynamical system with a saddle point equilibrium. By applying techniques developed for the problem of exit from a stable equilibrium, we obtain a new limit law for the exit time from a neighborhood of the saddle, assuming the initial point is on the stable manifold. The limit law shows that the exit distribution depends on (the logarithm of) the noise parameter in an additive way. This gives a more accurate description of the exit law than the previous (but more general) results of Kifer and Mikami. Generalization to higher dimensions seems likely, although only if the unstable manifold has dimension 1.