Limiting Exit Location Distributions in the Stochastic Exit Problem

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 ε, the system state will eventually leave the domain of attraction Ω of S. We analyze the case when, as ε→0, the exit location on the boundary ∂Ω 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 ∂Ω is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter μ, equal to the ratio |λₛ(H)|/λᵤ(H) of the stable and unstable eigenvalues of the linearized deterministic flow at H. If μ < 1, then the exit location distribution is generically asymptotic as ε→0 to a Weibull distribution with shape parameter 2/μ, on the O(εμ/2) lengthscale near H. If μ > 1, it is generically asymptotic to a distribution on the O(ε1/2) 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.