Abstract

Perturbation and approximation underly almost all applications of physical mathematical models. The averaging method, first introduced for approximate periodic motions, is now widely used for a large class of problems in both pure and applied mathematics. The purpose of averaging is easy to describe. Suppose that we have a system of variables interacting with each other and moving at different scales of speed (of order 1 and of order 1/epsilon) with the fast variables `fast oscillatory’. Both slow and fast variables evolve in time according to some rules, for example solving a family of differential or stochastic differential equations. The aim is to determine whether the slow variables can be approximated by an autonomous systems of equations, called the effective dynamic, as epsilon is taken to 0 and the speed of the fast variables tends to infinity. Slow/fast systems arise from perturbations of conservation laws and breaking of symmetries. We will discuss the latest developments in averaged dynamics, touching on recent work with M. Hairer on averaged dynamics with fractional noise (this leads to very different behaviour from the white noise case and requires new techniques).