Abstract:

The boundary of the domain of attraction of a dynamical system (e.g., a stable equilibrium, a limit cycle, etc.) is called a separatrix. The exit problem for a dynamical system driven by noise is to determine the probability distribution of points on the separatrix where the random trajectories of the noisy dynamics exit the domain of attraction. This problem is equivalent to that of finding the Green function for a Dirichlet problem inside the separatrix. Large Deviations Theory (LDT) predicts that in the limit of vanishing noise the distribution of the exit points is concentrated at the absolute minima of the action functional on the separatrix. We show that in Kramers' classical exit problem for small, but finite noise the exit points avoid the minimum of the action functional and we determine the asymptotic form of the exit distribution.

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