Asymptotic Exit Location Distributions in the Stochastic Exit Problem

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 $ε$, the system state will eventually leave the domain of attraction $Ω$ of $S$. We analyse the case when, as $ε\to0$, the exit location on the boundary $\partialΩ$ is increasingly concentrated near a saddle point $H$ of the deterministic dynamics. We show that the asymptotic form of the exit location distribution on $\partialΩ$ 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 $ε\to0$ to a Weibull distribution with shape parameter $2/μ$, on the $O(ε^{μ/2})$ length scale near $H$. If $μ>1$ it is generically asymptotic to a distribution on the $O(ε^{1/2})$ length scale, whose moments we compute. The asymmetry of the asymptotic exit location distribution is attributable to the generic presence of a `classically forbidden&#39; region: a wedge-shaped subset of $Ω$ with $H$ as vertex, which is reached from $S$, in the $ε\to0$ limit, only via `bent&#39; (non-smooth) fluctuational paths that first pass through the vicinity of $H$. We deduce from the presence of this forbidden region that the classical Eyring formula for the small-$ε$ exponential asymptotics of the mean first exit time is generically inapplicable.