Abstract. Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength, the system state will eventually leave the domain of attraction of. We analyse the case when, as, the exit location on the boundary is increasingly concentrated near a saddle point of the deterministic dynamics. We show using formal methods that the asymptotic form of the exit location distribution on is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter, equal to the ratio of the stable and unstable eigenvalues of the linearized deterministic flow at. If then the exit location distribution is generically asymptotic as! " to a Weibull distribution with shape parameter #$ % , on the &'(*) +-,. lengthscale near. If 0/1 it is generically asymptotic to a distribution on the &'(-23+, 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. Key words. Stochastic exit problem, large fluctuations, large deviations, Wentzell-Freidlin theory, exit location, saddle point avoidance, first passage time, matched asymptotic expansions, singular perturbation theory, stochastic analysis, Ackerberg-O’Malley resonance. AMS subject classifications. 60J60, 35B25, 34E20 1. Introduction. We

We consider the overdamped limit of two-dimensional double well systems perturbed by weak noise. In the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two wells. However, as the parameters of a symmetric double well system are varied, a unique MPEP may bifurcate into two equally likely MPEP's. At the bifurcation point in parameter space, the activation kinetics of the system become non-Arrhenius. In this paper we quantify the non-Arrhenius behavior of a system at the bifurcation point, by using the Maslov-WKB method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix. The approximation is a formal asymptotic solution of the Smoluchowski equation. Our analysis relies on the development of a new scaling theory, which yields `critical exponents' describing...

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 ffl, the system state will eventually leave the domain of attraction W of S. We analyse the case when, as ffl ! 0, the exit location on the boundary @W 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 @W is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter ¯, equal to the ratio j s (H)j= u (H) of the stable and unstable eigenvalues of the linearized deterministic flow at H . If ¯ ! 1 then the exit location distribution is generically asymptotic as ffl ! 0 to a Weibull distributionwith shape parameter 2=¯, on the O(ffl ¯=2 ) lengthscale near H . If ¯ ? 1 it is generically asymptotic to a distribution o...

We study the trajectories followed by a particle subjected to weak noise when escaping from the domain of attraction of a stable fixed point. If detailed balance is absent, a focus may occur along the most probable exit path, leading to a breakdown of symmetry (if present). The exit trajectory bifurcates, and the exit location distribution may become `skewed' (nonGaussian) . The weak-noise asymptotics of the mean escape time are strongly affected. Our methods extend to the study of skewed exit location distributions in stochastic models without symmetry. PACS numbers: 02.50.-r, 05.40.+j Partially supported by the National Science Foundation under grant NCR-90-16211. y Partially supported by the U.S. Department of Energy under grant DE-FG03-93ER25155. A particle moving in a force field, but weakly perturbed by external noise, will spend most of its time near stable fixed points of the force field. But the particle will occasionally undergo a large fluctuation: it will leave the b...