Limiting exit location distributions in the stochastic exit problem

Robert S Maier, Daniel L. Stein

Research output: Contribution to journalArticle

121 Citations (Scopus)

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 ∈, 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
Volume57
Issue number3
StatePublished - Jun 1997

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Exit Problem
Limiting
Formal methods
White noise
Byproducts
Exit Time
Dynamical systems
Matched Asymptotic Expansions
Random Perturbation
Stochastic Analysis
Domain of Attraction
Formal Methods
Saddlepoint
Limiting Distribution
Length Scale
Continuous Time
Dynamical system
Unstable
Fixed point
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Keywords

  • 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

  • Mathematics(all)
  • Applied Mathematics

Cite this

Limiting exit location distributions in the stochastic exit problem. / Maier, Robert S; Stein, Daniel L.

In: SIAM Journal on Applied Mathematics, Vol. 57, No. 3, 06.1997, p. 752-790.

Research output: Contribution to journalArticle

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