Averaging Principle For A Class Of Stochastic Differential Equations

This paper discusses the following slow-fast stochastic differential equations: where W, is one dimension standard Brownian motion. Averaging schemes were widely applied in the study of celestial mechanics. Highly oscillating components in a dynamical system may be "averaged out " under certain suitable conditions, from which an averaged, effective system emerges. Such an effective system is more amenable for analysis and simu-lation. Firstly, this paper shows that the solution u∈ of the slow system converges in probabil-ity to the solution u of its averaging equation du= (-4u-u3)dt, and the order of convergence is (?)∈that is u∈-u= O((?)∈). Secondly, using martingale representation method, it is proved that the deviation of the slow system and the averaging system converges in distribution to a Gauss process, that is, z is a Gauss process.