Additive white noise may significantly increase the response of bistable systems to a periodic driving signal. We consider two classes of double-well potentials, symmetric and asymmetric, modulated periodically in time with period 1/, where is a moderately (not exponentially) small parameter. We show that the response of the system changes drastically when the noise intensity σ crosses a threshold value. Below the threshold, paths are concentrated near one potential well, and have an

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... ly 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 localised in time near the points of minimal barrier height. We give a mathematically rigorous description of the behaviour of individual paths, which allows us, in particular, to determine the power-law dependence of the critical noise intensity on and on the minimal barrier height, as well as the asymptotics of the transition and non-transition probabilities.