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research
Asymptotic Exit Location Distributions in the Stochastic Exit Problem
Authors
Ackerberg R. C.
Daniel L. Stein
Day M. V.
Robert S. Maier
Publication date
28 July 1994
Publisher
'Society for Industrial & Applied Mathematics (SIAM)'
Doi
Cite
View
on
arXiv
Abstract
Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point
S
S
S
. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength
ϵ
\epsilon
ϵ
, the system state will eventually leave the domain of attraction
Ω
\Omega
Ω
of
S
S
S
. We analyse the case when, as
ϵ
→
0
\epsilon\to0
ϵ
→
0
, the exit location on the boundary
∂
Ω
\partial\Omega
∂
Ω
is increasingly concentrated near a saddle point
H
H
H
of the deterministic dynamics. We show that the asymptotic form of the exit location distribution on
∂
Ω
\partial\Omega
∂
Ω
is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter
μ
\mu
μ
, equal to the ratio
∣
λ
s
(
H
)
∣
/
λ
u
(
H
)
|\lambda_s(H)|/\lambda_u(H)
∣
λ
s
​
(
H
)
∣/
λ
u
​
(
H
)
of the stable and unstable eigenvalues of the linearized deterministic flow at
H
H
H
. If
μ
<
1
\mu<1
μ
<
1
then the exit location distribution is generically asymptotic as
ϵ
→
0
\epsilon\to0
ϵ
→
0
to a Weibull distribution with shape parameter
2
/
μ
2/\mu
2/
μ
, on the
O
(
ϵ
μ
/
2
)
O(\epsilon^{\mu/2})
O
(
ϵ
μ
/2
)
length scale near
H
H
H
. If
μ
>
1
\mu>1
μ
>
1
it is generically asymptotic to a distribution on the
O
(
ϵ
1
/
2
)
O(\epsilon^{1/2})
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
\Omega
Ω
with
H
H
H
as vertex, which is reached from
S
S
S
, in the
ϵ
→
0
\epsilon\to0
ϵ
→
0
limit, only via `bent' (non-smooth) fluctuational paths that first pass through the vicinity of
H
H
H
. We deduce from the presence of this forbidden region that the classical Eyring formula for the small-
ϵ
\epsilon
ϵ
exponential asymptotics of the mean first exit time is generically inapplicable.Comment: This is a 72-page Postscript file, about 600K in length. Hardcopy requests to
[email protected]
or
[email protected]
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Last time updated on 05/06/2019