Slow manifold and averaging for slow–fast stochastic differential system
Slow manifold and averaging for slow–fast stochastic differential system
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DOI：
10.1016/j.jmaa.2012.09.029
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发表时间：
2009-03
影响因子：
1.3
通讯作者：
Wei Wang;Wei Wang;Anthony J. Roberts
中科院分区：
文献类型：
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作者：
Wei Wang;Wei Wang;Anthony J. Roberts
我们考虑具有多个时间尺度的随机动力系统。当快速模式由白噪声驱动时，获得并探索了慢-快系统的中间简化模型。首先，为了后面的比较，指数吸引随机慢流形上的简化随机系统的一个近似被导出为 O(ϵ) 阶误差。其次，由于噪声仅驱动快速模式，因此求平均会得出误差为 O(ϵ) 阶的自主确定性系统。然后，鞅论证解释了平均系统的波动，以形成一个误差为 O(ϵ) 阶的中间简化模型——现在由白噪声驱动的自主确定性系统。该中间简化模型比随机慢流形上的简化模型具有更简单的形式。这些结果不仅将平均与随机慢流形联系起来，还提供了一种改进随机系统平均模型的鞅方法。
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). 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)—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.