We consider a random perturbation of a pseudoperiodic flow on $\R^2$. The structure of such flows has been studied by Arnol'd; it contains regions where there are local Hamiltonians, and an ergodic region. Under an appropriate change of time, we identify a reduced model as the strength of the random perturbation tends to zero (along a certain subsequence). In the Hamiltonian region, arguments of Freidlin and Wentzell are used to identify a limiting graph-valued process. The ergodic region is reduced to a single point, which is "sticky". The identification of the glueing conditions which rigorously describe this stickiness follows from a perturbed test-function analysis in the ergodic region.
"Random perturbations of two-dimensional pseudoperiodic flows." Illinois J. Math. 50 (1-4) 853 - 959, 2006. https://doi.org/10.1215/ijm/1258059495