- 00-517 Nils Berglund and Barbara Gentz
- A sample-paths approach to noise-induced synchronization:
Stochastic resonance in a double-well potential
Dec 29, 00
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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.