A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential

Additive white noise may significantly increase the response of bistable systems to a periodic driving signal. We consider the overdamped motion of a Brownian particle in two classes of double-well potentials, symmetric and asymmetric ones. These potentials are modulated periodically in time with period 1/\eps, where \eps is a moderately (not exponentially) small parameter. We show that the response of the system changes drastically when the noise intensity . crosses a threshold value. Below the threshold, paths are concentrated in one potential well, and have an exponentially small probability to jump to the other well. Above the threshold, transitions between the wells occur with probability exponentially close to 1/2 in the symmetric case, and exponentially close to 1 in the asymmetric case. The transition zones are localized in time near the instants of minimal barrier height. We give a mathematically rigorous description of the behavior of individual paths, which allows us, in particular, to determine the power-law dependence of the critical noise intensity on \eps and on the minimal barrier height, as well as the asymptotics of the transition and nontransition probabilities.