L2 Diffusion Approximation for Slow Motion in Averaging

Abstract

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.