Nils Berglund and Barbara Gentz
A sample-paths approach to noise-induced synchronization: 
Stochastic resonance in a double-well potential
(847K, PostScript)

ABSTRACT.  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.