Asymptotic Exit Location Distributions in the Stochastic Exit Problem
| dc.creator | Maier, Robert S. | |
| dc.creator | Stein, Daniel L. | |
| dc.date | 1994-07-28 | |
| dc.date.accessioned | 2026-07-07T09:05:29Z | |
| dc.date.available | 2026-07-07T09:05:29Z | |
| dc.description | 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 $ε$, the system state will eventually leave the domain of attraction $Ω$ of $S$. We analyse the case when, as $ε\to0$, the exit location on the boundary $\partialΩ$ is increasingly concentrated near a saddle point $H$ of the deterministic dynamics. We show that the asymptotic form of the exit location distribution on $\partialΩ$ 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 $ε\to0$ to a Weibull distribution with shape parameter $2/μ$, on the $O(ε^{μ/2})$ length scale near $H$. If $μ>1$ it is generically asymptotic to a distribution on the $O(ε^{1/2})$ length scale, whose moments we compute. The asymmetry of the asymptotic exit location distribution is attributable to the generic presence of a `classically forbidden' region: a wedge-shaped subset of $Ω$ with $H$ as vertex, which is reached from $S$, in the $ε\to0$ limit, only via `bent' (non-smooth) fluctuational paths that first pass through the vicinity of $H$. We deduce from the presence of this forbidden region that the classical Eyring formula for the small-$ε$ exponential asymptotics of the mean first exit time is generically inapplicable. | |
| dc.description | This is a 72-page Postscript file, about 600K in length. Hardcopy requests to rsm@math.arizona.edu or dls@ccit.arizona.edu | |
| dc.identifier | https://arxiv.org/abs/adap-org/9407003 | |
| dc.identifier | http://arxiv.org/abs/adap-org/9407003 | |
| dc.identifier | SIAM J. Appl. Math. 57 (1997) 752 | |
| dc.identifier | doi:10.1137/S0036139994271753 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/149693 | |
| dc.subject | Adaptation and Self-Organizing Systems | |
| dc.subject | Chaotic Dynamics | |
| dc.title | Asymptotic Exit Location Distributions in the Stochastic Exit Problem | |
| dc.type | text |