### 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 |λ_{s}(H)|/λ_{u}(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 language | English (US) |
---|---|

Pages (from-to) | 752-790 |

Number of pages | 39 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 57 |

Issue number | 3 |

State | Published - Jun 1997 |

### Fingerprint

### 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

*SIAM Journal on Applied Mathematics*,

*57*(3), 752-790.

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

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 57, no. 3, pp. 752-790.

}

TY - JOUR

T1 - Limiting exit location distributions in the stochastic exit problem

AU - Maier, Robert S.

AU - Stein, Daniel L.

PY - 1997/6

Y1 - 1997/6

N2 - 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.

AB - 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.

KW - Ackerberg - O'malley resonance

KW - Exit location

KW - First passage time

KW - Large deviations

KW - Large fluctuations

KW - Matched asymptotic expansions

KW - Saddle point avoidance

KW - Singular perturbation theory

KW - Stochastic analysis

KW - Stochastic exit problem

KW - Wentzell-Freidlin theory

UR - http://www.scopus.com/inward/record.url?scp=0031162897&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031162897&partnerID=8YFLogxK

M3 - Article

VL - 57

SP - 752

EP - 790

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 3

ER -