Slow manifold and averaging for slow-fast stochastic differential system

Abstract

We consider multiscale stochastic dynamical systems. In this article an \emph{intermediate} reduced model is obtained for a slow-fast system with fast mode driven by white noise. First, the reduced stochastic system on exponentially attracting slow manifold reduced system is derived to errors of $\mathcal{O}(\epsilon)$. Second, averaging derives an autonomous deterministic system up to errors of $\mathcal{O}(\sqrt{\epsilon})$. Then an intermediate reduced model, which is an autonomous deterministic system driven by white noise up to errors of $\mathcal{O}(\epsilon)$, is derived using a martingale approach to account for fluctuations about the averaged system. This intermediate reduced model has a simpler form than the reduced model on the stochastic slow manifold.