Surmounting fluctuating barriers: A simple model in discrete time

The escape rate is calculated for a piecewise linear map in the presence of two additive weak noises: Gaussian white noise (thermal noise) and Ornstein-Uhlenbeck noise (barrier fluctuations). A transition state theory yields the exact escape rate in the weak noise limit. By including the dominant finite-noise corrections we find very good agreement with numerical simulations. Of particular interest is the behavior of the escape rate k({τ}) as a function of the correlation time {τ} of the Ornstein-Uhlenbeck noise. We show that the qualitative behavior of k({τ}) at any fixed thermal noise intensity is the same as in the absence of thermal noise. Whether k({τ}) exhibits a local maximum (resonant activation) is determined by the detailed {τ} dependence of the colored-noise intensity: e.g., resonant activation is found (at a correlation time of order 1) if the integral autocorrelation of the colored noise is kept {τ} independent but not if the variance is kept {τ} independent. In the continuous-time limit neither of these cases shows resonant activation.