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Limiting exit location distributions in the stochastic exit problem
 SIAM J. Appl. Math
, 1997
"... Abstract. Consider a twodimensional continuoustime 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 ..."
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Cited by 39 (1 self)
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Abstract. Consider a twodimensional continuoustime 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 nonGaussian 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 weaknoise exit time asymptotics. Key words. Stochastic exit problem, large fluctuations, large deviations, WentzellFreidlin theory, exit location, saddle point avoidance, first passage time, matched asymptotic expansions, singular perturbation theory, stochastic analysis, AckerbergO’Malley resonance. AMS subject classifications. 60J60, 35B25, 34E20 1. Introduction. We
A Scaling Theory of Bifurcations in the Symmetric WeakNoise Escape Problem
, 1995
"... We consider the overdamped limit of twodimensional 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 we ..."
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Cited by 12 (1 self)
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We consider the overdamped limit of twodimensional 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 nonArrhenius. We quantify the nonArrhenius behavior of a system at the bifurcation point, by using the MaslovWKB 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 construction of a new scaling theory, which yields ‘critical exponents’ describing weaknoise behavior at the bifurcation point, near the saddle.
c © World Scientic Publishing Company LARGE FLUCTUATIONS IN A PERIODICALLY DRIVEN DYNAMICAL SYSTEM
, 1996
"... Fluctuations in a periodically driven overdamped oscillator are studied theoretically and experimentally in the limit of low noise intensity by investigation of their prehistory. It is shown that, for small noise intensity, fluctuations to points in coordinate space that are remote from the stable ..."
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Fluctuations in a periodically driven overdamped oscillator are studied theoretically and experimentally in the limit of low noise intensity by investigation of their prehistory. It is shown that, for small noise intensity, fluctuations to points in coordinate space that are remote from the stable states occur along paths that form narrow tubes. The tubes are centered on the optimal paths corresponding to trajectories of an auxiliary Hamiltonian system. The optimal paths themselves, and the tubes of paths around them, are visualized through measurements of the prehistory probability distribution for an electronic model. Some general features of fluctuations in nonequilibrium systems, such as singularities in the pattern of optimal paths, the corresponding nondierentiability of the generalized nonequilibrium potential, and the feasibility of their experimental investigation, are discussed. 1.
Freddy Bouchet, Julien Reygner. Generalisation of the EyringKramers transition rate formula
, 2015
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