$\epsilon$ should thus reveal the qualitative structure of the unperturbed potential $V_0$. In this paper we test this paradigm by studying a two dimensional example with two wells and two independent pathways between them, see Section \[sum\_chap\_mexican\_hat\_toy\_model\].

Historical Background
---------------------

Stochastic resonance has attracted interest among mathematicians and physicist. An overview of the studies that have occurred in both physics and mathematics are given here.

### Physical Background

Stochastic resonance was first observed in 1981 [@benzi81; @tk_17c; @tk_17b]. The first example [@benzi81] considered transitions between two metastable states to model the cyclic occurrences of ice ages. Since then many examples of stochastic resonance were found in optics [@opt_1_PhysRevLett.60.2626; @opt_2_Guidoni1995; @opt_3_PhysRevA.49.2199; @opt_4_PhysRevLett.68.3375], electronics [@elec_1_FAUVE19835; @elect_2_PhysRevE.49.R1792; @elect_3_Mantegna1995; @elect_4_PhysRevLett.76.563; @elect_5_PhysRevLett.74.3161; @elect_6_:/content/aip/journal/jap/76/10/10.1063/1.358258; @elect_7_PhysRevA.39.4323; @elect_8_PhysRevE.49.4878; @elect_9_PhysRevLett.67.1799], neuronal systems [@neuron_1_PhysRevLett.67.656], quantum systems [@quant_1_:/content/aip/journal/jap/77/6/10.1063/1.358720; @quant_2_:/content/aip/journal/apl/66/1/10.1063/1.114161] and paddlefish [@fish_1_PhysRevLett.84.4773; @fish_2_FREUND200271]. Stochastic resonance could be thought of as quasi-deterministically periodic transition between two metastable states. For example, the climate of the Earth could be modelled by two states. There is a state corresponding to an Ice Age and another corresponding to the opposite of an Ice Age, a so-called “Hot Age”. As the Earth’s climate cyclically changes many times between Cold Ages and Hot Ages, its behaviour could be 