Colored noise and a characteristic level crossing problem

In this paper we discuss the activation of a dynamical system of one spatial variable forced by an external Gaussian stationary colored noise. The model is an analog of the Smoluchowski equation when external fluctuations are present. We assume that the underlying deterministic dynamics are derived from a potential forming a well. We enlarge the state space by including the noise variable, and then use half-range expansion techniques, singular perturbations, and matched asymptotics to solve the boundary value problem for the steady-state probability density function, in the asymptotic limit of small correlation time and over the entire regime of noise strength. We find a uniformly valid expression for the activation rate to the top of the potential barrier (a characteristic point for the unforced dynamical system). We show the important effects of the boundary behavior of the process on its activation rate. If the intensity of the noise is small compared to the autocorrelation of the noise, then the effective activation energy is modified by the noise autocorrelation. We consider an example of the activation process in a double well potential, and compare our uniformly valid results with results reported in the literature.

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