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 $\epsilon$, the system state will eventually leave the domain of attraction $\Omega$ of $S$. We analyse the case when, as $\epsilon \to 0$, the exit location on the boundary $\partial \Omega$ is increasingly concentrated near a saddle point $H$ of the deterministic dynamics. We show 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 $\epsilon \to 0$ to a Weibull distribution with shape parameter $2/\mu$, on the $O({\epsilon }^{\mu /2})$ length scale near $H$. If $\mu >1$ it is generically asymptotic to a distribution on the $O({\epsilon }^{1/2})$ length scale, whose moments we compute. The asymmetry of the asymptotic exit location distribution is attributable to the generic presence of a `classically forbidden' region: a wedge-shaped subset of $\Omega$ with $H$ as vertex, which is reached from $S$, in the $\epsilon \to 0$ limit, only via `bent' (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-$\epsilon$ exponential asymptotics of the mean first exit time is generically inapplicable.

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References in corpus (5)

• The Escape Problem for Irreversible Systems
• A Scaling Theory of Bifurcations in the Symmetric Weak-Noise Escape Problem
• Observable and hidden singular features of large fluctuations in nonequilibrium systems
• The Effect of Focusing and Caustics on Exit Phenomena in Systems Lacking Detailed Balance
• Large Fluctuations in Stochastically Perturbed Nonlinear Systems: Applications in Computing

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