Limiting Exit Location Distributions in the Stochastic Exit Problem

PROBLEM ROBERT S. MAIER AND DANIEL L. STEINy Abstract. Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point S. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength , the system state will eventually leave the domain of attraction of S. We analyse the case when, as ! 0, the exit location on the boundary @ is increasingly concentrated near a saddle point H of the deterministic dynamics. We show using formal methods that the asymptotic form of the exit location distribution on @ is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter , equal to the ratio j s(H)j= u(H) of the stable and unstable eigenvalues of the linearized deterministic flow at H. If < 1 then the exit location distribution is generically asymptotic as ! 0 to a Weibull distribution with shape parameter 2= , on the O( =2) lengthscale near H. If > 1 it is generically asymptotic to a distribution on the O( 1=2) lengthscale, whose moments we compute. Our treatment employs both matched asymptotic expansions and stochastic analysis. As a byproduct of our treatment, we clarify the limitations of the traditional Eyring formula for the weak-noise exit time asymptotics.