Limiting exit location distributions in the stochastic exit problem

Robert S. Maier, Daniel L. Stein

    Research output: Contribution to journalArticle

    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
    Classify

    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

    Maier, Robert S. ; Stein, Daniel L. / Limiting exit location distributions in the stochastic exit problem. In: SIAM Journal on Applied Mathematics. 1997 ; Vol. 57, No. 3. pp. 752-790.
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