Robert S. Maier and Daniel L. Stein
Asymptotic Exit Location Distributions in the Stochastic Exit Problem
(606K, Postscript [72 pages])
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 $\epsilon$, the system state
will eventually leave the domain of attraction $\Omega$ of $S$.
We analyse the case when, as $\epsilon\to0$, the exit location on the
boundary $\partial\Omega$ is 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 $\partial\Omega$
is generically non-Gaussian and asymmetric, and classify the possible
limiting distributions. A key role is played by a parameter $\mu$, equal
to the ratio $|\lambda_s(H)|/\lambda_u(H)$ of the stable and unstable
eigenvalues of the linearized deterministic flow at $H$. If $\mu<1$ then
the exit location distribution is generically asymptotic as $\epsilon\to0$
to a Weibull distribution with shape parameter $2/\mu$, on the
$O(\epsilon^{\mu/2})$ lengthscale near $H$. If $\mu>1$ it is generically
asymptotic to a distribution on the $O(\epsilon^{1/2})$ lengthscale, 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 $\Omega$ with $H$ as vertex,
which is reached from $S$, in the $\epsilon\to0$ limit, only via `bent'
(non-smooth) fluctuational paths that first pass through the vicinity
of $H$. We show that as a consequence the Wentzell-Freidlin quasipotential
function $W$, which governs the frequency of fluctuations to the vicinity
of any point $\vec x$ in $\Omega$ and is the solution of a Hamilton-Jacobi
equation, generically fails to be twice differentiable at ${\vec x}=H$.
This nondifferentiability implies that the classical Eyring formula for the
small-$\epsilon$ exponential asymptotics of the mean first exit time, which
includes a prefactor involving the Hessian of $W$ at ${\vec x}=H$, is
generically inapplicable. Our treatment employs both matched asymptotic
expansions and probabilistic analysis. Besides relating our results to
the work of others on the stochastic exit problem, we comment on their
implication for the two-dimensional analogue of Ackerberg-O'Malley
resonance.