L2 Diffusion Approximation for Slow Motion in Averaging

Assuming that the fast motion in averaging is sufficiently well mixing we show that the slow motion can be approximated in the L2-sense by a diffusion solving Hasselmann's nonlinear stochastic differential equation and which provides a much better approximation than the one suggested by the averaging principle. Previously, only weak limit theorems in averaging were known which cannot justify, in principle, a nonlinear diffusion approximation of the slow motion.