Boundary Value Problems for Statistics of Diffusion in a Randomly Switching Environment: PDE and SDE Perspectives
Boundary Value Problems for Statistics of Diffusion in a Randomly Switching Environment: PDE and SDE Perspectives
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随机切换环境中扩散统计的边值问题：PDE 和 SDE 视角
DOI：
10.1137/15m1038426
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
S. Lawley
中科院分区：
文献类型：
--
作者：
S. Lawley
受不同应用的驱动，最近的几个模型对PDE或PDE施加随机切换的边界条件。本文的目的是提供工具来计算这些模型的统计数据，并建立在随机环境中的扩散这两个角度之间的连接。在一般条件下，我们证明了随机切换偏微分方程的解的矩满足一个层次的BVP低阶矩耦合到高阶矩的边界。此外，我们证明了一组粒子的联合出口统计随机切换后，满足相应的层次结构的BVP。特别是，$M$时刻的解决方案切换偏微分方程对应于退出统计的$M$粒子后切换的偏微分方程。我们注意到，尽管粒子之间没有相互作用，但它们仍然是相关的，因为它们都遵循相同的开关模式。最后，我们给出了几个例子，我们的定理如何揭示有时令人惊讶的动态…
Driven by diverse applications, several recent models impose randomly switching boundary conditions on either a PDE or SDE. The purpose of this paper is to provide tools for calculating statistics of these models and establish a connection between these two perspectives on diffusion in a random environment. Under general conditions, we prove that the moments of a solution to a randomly switching PDE satisfy a hierarchy of BVPs with lower order moments coupling to higher order moments at the boundaries. Further, we prove that joint exit statistics for a set of particles following a randomly switching SDE satisfy a corresponding hierarchy of BVPs. In particular, the $M$th moment of a solution to a switching PDE corresponds to exit statistics for $M$ particles following a switching SDE. We note that though the particles are noninteracting, they are nonetheless correlated because they all follow the same switching SDE. Finally, we give several examples of how our theorems reveal the sometimes surprising dynam...