94-245 Robert S. Maier and Daniel L. Stein
Asymptotic Exit Location Distributions in the Stochastic Exit Problem (606K, Postscript [72 pages]) Jul 27, 94
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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.

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