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 ∈, the system state will eventually leave the domain of attraction Ωof S. We analyze the case when, as ∈ → 0, the exit location on the boundary ∂Ωis 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 |λ_s(H)|/λ_u(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 ∈ → 0 to a Weibull distribution with shape parameter 2/μ, on the script O sign(∈^μ/2) lengthscale near H. If μ > 1, it is generically asymptotic to a distribution on the script O sign(∈^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.