Limiting exit location distributions in the stochastic exit problem

Robert S. Maier, Daniel L. Stein

Research output: Contribution to journalArticlepeer-review

154 Scopus citations


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 ∈, the system state will eventually leave the domain of attraction Ωof S. We analyze the case when, as ∈ → 0, the exit location on the boundary ∂Ωis 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 ∂Ω 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 |λs(H)|/λu(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 ∈ → 0 to a Weibull distribution with shape parameter 2/μ, on the script O sign(∈μ/2) lengthscale near H. If μ > 1, it is generically asymptotic to a distribution on the script O sign(∈1/2) 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.

Original languageEnglish (US)
Pages (from-to)752-790
Number of pages39
JournalSIAM Journal on Applied Mathematics
Issue number3
StatePublished - Jun 1997


  • Ackerberg - O'malley resonance
  • Exit location
  • First passage time
  • Large deviations
  • Large fluctuations
  • Matched asymptotic expansions
  • Saddle point avoidance
  • Singular perturbation theory
  • Stochastic analysis
  • Stochastic exit problem
  • Wentzell-Freidlin theory

ASJC Scopus subject areas

  • Applied Mathematics


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