  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 $1/\eps$, where $\eps$ is a moderately (not exponentially) small
parameter. We show that the response of the system changes drastically when the
noise intensity $\sigma$ crosses a threshold value. Below the threshold, paths
are concentrated near 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 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 $\eps$ and on the minimal barrier height, as well as the
asymptotics of the transition and non-transition probabilities.
