The next order of approximation was rigorously derived by Bovier, Eckhoff, Gayrard, Klein [@Bovier02metastabilityin] and Berglund and Gentz [@kram_2_2008arXiv0807.1681B] using techniques from potential theory. Berglund and Gentz in a series of papers studied the situation of low, non-quadratic barriers and drifts not given by autonomous potentials [@kram_2_2008arXiv0807.1681B; @tk_16]. A review of different techniques used to derive Kramers’ formula can be found in the review paper [@krammer_review_2011arXiv1106.5799B].

In [@ld_1_comp_1_Freidlin2000333] Friedlin considered stochastic resonance in the adiabatic regime. This means the diffusion can effectively be described by a Markov process which describes the jumps between wells. This problem was revisited by Hermann, Imkeller and Pavlyukevich, see Chapter 4 in [@tran2014] and references therein, to derive results uniformly for varying time scale to identify the optimal resonance point asymptotically for small noise even outside the adiabatic regime leading to different logarithmic corrections including the famous cycling effect discovered by Day [@tk_20], see also [@tk_18] for the connection with stochastic resonance. Escape time outside of adiabatic regime is studied in [@tk_00_doi:10.1137/120887965].

As mentioned above in leading order the transitions of the diffusion process $X^\epsilon_t$ between the wells can be approximated by a two state Markov Chain $Y^\epsilon_t=\pm1$ which have been studied [@pav_thesis; @pav_imkell; @pav03; 