A boundary layer theory for diffusively perturbed transport around a heteroclinic cycle

We consider a two-dimensional transport equation subject to small diffusive perturbations. The transport equation is given by a Hamiltonian flow near a compact and connected heteroclinic cycle. We investigate approximately harmonic functions corresponding to the generator of the perturbed transport equation. In particular, we investigate such functions in the boundary layer near the heteroclinic cycle; the space of these functions gives information about the likelihood of a particle moving a mesoscopic distance into one of the regions where the transport equation corresponds to periodic oscillations (i.e., a "well" of the Hamiltonian). We find that we can construct such approximately harmonic functions (which can be used as "corrector functions" in certain averaging questions) when certain macroscopic "gluing conditions" are satisfied. This provides a different perspective on some previous work of Freidlin and Wentzell on stochastic averaging of Hamiltonian systems.