Averaging of stochastic flows: Twist maps and escape from resonance

Our setup is a classical stochastic averaging one studied by Has’minskiĭ, which is a two-dimensional SDE (on a cylinder) consisting of a fast angular drift and a slow axial diffusion. We seek to understand the asymptotics of the flow generated by this SDE. To do so, we fix nn initial points on the cylinder and consider the axial components of the trajectories evolving from these points. We conclude a propagation-of-chaos. There are two components of the limiting nn-point motion: a common Brownian motion, and nn independent Brownian motions, one for each initial point.