Asymptotic Exit Location Distributions in the Stochastic Exit Problem
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
ϵ
, the system state will eventually leave the domain of attraction
Ω
of
S
. We analyse the case when, as
ϵ→0
, the exit location on the boundary
∂Ω
is increasingly concentrated near a saddle point
H
of the deterministic dynamics. We show 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
ϵ→0
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→0
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.