Averaging of stochastic flows: Twist maps and escape from resonance

Our setup is a classical stochastic averaging one studied by Has'minskiǐ, 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 n 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 n-point motion: a common Brownian motion, and n independent Brownian motions, one for each initial point.