Rs. Maier et Dl. Stein, LIMITING EXIT LOCATION DISTRIBUTIONS IN THE STOCHASTIC EXIT PROBLEM, SIAM journal on applied mathematics, 57(3), 1997, pp. 752-790
Consider a two-dimensional continuous-time dynamical system, with an a
ttracting fixed point S. If the deterministic dynamics are perturbed b
y white noise (random perturbations) of strength epsilon, the system s
tate will eventually leave the domain of attraction Omega of S. We ana
lyze the case when, as epsilon --> 0, the exit location on the boundar
y partial derivative Omega is increasingly concentrated near a saddle
point H of the deterministic dynamics. We show using formal methods th
at the asymptotic form of the exit location distribution on partial de
rivative Omega is generically non-Gaussian and asymmetric, and classif
y the possible limiting distributions. A key role is played by a param
eter mu, equal to the ratio \lambda(s)(H)\lambda(u)(H) of the stable a
nd unstable eigenvalues of the linearized deterministic flow at H. If
mu < 1, then the exit location distribution is generically asymptotic
as epsilon --> 0 to a Weibull distribution with shape parameter 2/mu,
on the O(epsilon(mu/2)) lengthscale near H, If mu > 1, it is generical
ly asymptotic to a distribution on the O(epsilon(1/2)) lengthscale, wh
ose moments we compute. Our treatment employs both matched asymptotic
expansions and stochastic analysis. As a byproduct of our treatment, w
e clarify the limitations of the traditional Eyring formula for the we
ak-noise exit time asymptotics.