LIMITING EXIT LOCATION DISTRIBUTIONS IN THE STOCHASTIC EXIT PROBLEM

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.