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 $\epsilon$, the system state will eventually leave the domain of attraction $\Omega$ of S. We analyze the case when, as $\epsilon\to0$, the exit location on the boundary $\partial\Omega$ 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 $\partial\Omega$ is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter $\mu$, equal to the ratio $|\lambda_s(H)|/\lambda_u(H)$ of the stable and unstable eigenvalues of the linearized deterministic flow at H. If $\mu < 1$, then the exit location distribution is generically asymptotic as $\epsilon\to0$ to a Weibull distribution with shape parameter $2/\mu$, on the ${\cal O}(\epsilon^{\mu/2})$ ...