Slow manifold and averaging for slow–fast stochastic differential system

We consider stochastic dynamical systems with multiple time scales. An intermediate reduced model is obtained and explored for a slow–fast system when the fast mode is driven by white noise. First, and for later comparison, one approximation to the reduced stochastic system on the exponentially attracting stochastic slow manifold is derived to errors of order O(ϵ)O(ϵ). Second, because the noise only drives the fast modes, averaging derives an autonomous deterministic system with errors of order O(ϵ). Then a martingale argument accounts for fluctuations about the averaged system to form an intermediate reduced model with errors of order O(ϵ)O(ϵ)—the autonomous deterministic system now driven by white noise. This intermediate reduced model has a simpler form than the reduced model on the stochastic slow manifold. These results not only connect averaging with the stochastic slow manifold, they also provide a martingale method for improving averaged models of stochastic systems.