modelled by stochastic resonance.

A range of techniques for example linear response [@lin_1_TUS:TUS1787; @lin_2_doi:10.1137/0143037], signal-to-noise ratio [@sign_1_PhysRevA.39.4668; @sign_2_escape_2_PhysRevA.41.4255] and distribution of escape times [@escape_1_PhysRevA.42.3161; @sign_2_escape_2_PhysRevA.41.4255; @escape_3_PhysRevE.49.4821] were used to define, analyse and study stochastic resonance. These techniques along with other examples of stochastic resonance are reviewed in the long overview paper by Gammaitoni, Hänggi, Jung and Marchesoni [@RevModPhys.70.223]. We will evaluate the usefulness of some of these techniques for our problem, see Section \[sum\_chap\_results\].

### Mathematical Background

There are various mathematical studies of stochastic resonance. These often involve different orders of approximations for small noise levels. The first and second order of approximations are discussed below. Adiabatic large deviation is also presented.

In the first leading order of approximation, a key element of study is to control the escape times from the wells as given by the so called large deviation theory, see the monograph of Freidlin and Wentzell [@freidlin98]. The distribution of the exit time was derived by Day in [@tk_19] and by Galves, Kifer, Olivieri and Vares [@freid_2_doi:10.1137/1119057; @oliveri_expoen; @freid_3]. To go beyond leading order has been much more difficult for the transition problem between two wells as WKB theory could up to now not be rigorously applied.

