A transition in a noisy linear system driven by a periodic signal

We consider a one-dimension linear random walk between two trapping points in which the transition probabilities vary periodically in time. An earlier analysis of this system showed that the mean time to trapping of a particle in this system exhibits a minimum when considered as a function of frequency. In this note we show that this parameter makes a transition in behavior from a monotonic decrease with increasing amplitude of the periodic term to a monotonic increase with this parameter depending on the frequency. A physical argument is suggested to explain this behavior. Confirmation of this crossover can also be derived from a diffusion model.