Two-time-scale stochastic partial differential equations driven by $\alpha $-stable noises: Averaging principles
Two-time-scale stochastic partial differential equations driven by $\alpha $-stable noises: Averaging principles
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DOI：
10.3150/14-bej677
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发表时间：
2016-09
期刊：
影响因子：
1.5
通讯作者：
J. Bao;G. Yin;C. Yuan
中科院分区：
文献类型：
--
作者：
J. Bao;G. Yin;C. Yuan
本文主要研究双时间尺度下的随机偏微分方程组。与现有文献中的工作不同，系统是由$\α$-稳定过程驱动的，其中$\α\in(1，2)$。此外，SPDEs要么由具有有限状态空间的连续时间马尔可夫链调制，要么具有附加的快速跳跃分量。马尔可夫链的加入是为了处理随机环境的需要，而快跳过程的加入使得能够考虑快过程样本路径的不连续性。在假设一个快速变化的马尔可夫切换或一个附加的快速变化跳跃过程的情况下，本工作的目的是得到这类系统的平均原理。有几个明显的困难。首先，噪声不是平方可积的。其次，在我们的方案中，对于基本的SPDE，只有一个唯一的温和解，因此只有温和的S公式可用.此外，另一个新的方面是在公式中加入了快速区域切换和快速变化的跳跃过程，这扩大了基本系统的适用性.为了克服这些困难，采用了半群方法.在适当的条件下，证明了$p$阶矩收敛发生在$p\in(1，α)$，这比通常的弱收敛方法更强.
This paper focuses on stochastic partial differential equations (SPDEs) under two-time-scale formulation. Distinct from the work in the existing literature, the systems are driven by $\alpha$-stable processes with $\alpha \in(1,2)$. In addition, the SPDEs are either modulated by a continuous-time Markov chain with a finite state space or have an addition fast jump component. The inclusion of the Markov chain is for the needs of treating random environment, whereas the addition of the fast jump process enables the consideration of discontinuity in the sample paths of the fast processes. Assuming either a fast changing Markov switching or an additional fast-varying jump process, this work aims to obtain the averaging principles for such systems. There are several distinct difficulties. First, the noise is not square integrable. Second, in our setup, for the underlying SPDE, there is only a unique mild solution and as a result, there is only mild It\^{o}'s formula that can be used. Moreover, another new aspect is the addition of the fast regime switching and the addition of the fast varying jump processes in the formulation, which enlarges the applicability of the underlying systems. To overcome these difficulties, a semigroup approach is taken. Under suitable conditions, it is proved that the $p$th moment convergence takes place with $p\in(1,\alpha )$, which is stronger than the usual weak convergence approaches.