Lévy噪声随机偏微分方程的平均原理

The problem of model reduction is one of the central problems in multiscale models that have recently attracted much attention from the research community across a number of disciplines ranging from applied mathematics to physics, biology and engineering. This project will focus on averaging principle for a class of stochastic partial differential equations (SPDEs) with fast and slow time scales, which are driven by Lévy processes. The effects of large scale and random influence are two main difficulties in solving this problem. With the averaging principle and stochastic analysis, we will study the necessary conditions so as that there exist a reduction equation which approximates the dominant component of the SPDEs with two time scales, the explicit order of convergence (with respect to the parameter of time scale) in strong sense (approximation of trajectories) and in weak sense (approximation of laws) for the approximation of dominant component towards the solution of this reduction equation. The theoretical significance of our research will make progress in understanding the evolutionary behavior for stochastic systems with multiple scales. It also provides a rigorous theoretical basis for modeling, simulation, parameter estimation, optimal control for complex systems with multiple scales.

多尺度模型正引起数学、物理、生物和工程等诸多学科的极大关注。解决多尺度模型的核心问题之一就是对模型进行约化。本项目中，我们将研究Lévy过程驱动的、具有快慢两个时间尺度的随机偏微分方程的平均化问题。大尺度效应和随机影响是解决该问题的两个主要难点。在平均化原理和随机分析理论的框架下，我们将研究两时间尺度的随机偏微分方程的约化方程存在以及逼近原系统主要分量的必要条件、强收敛(轨道的逼近)及弱收敛(分布的逼近)意义下，主要分量与约化方程的解过程关于时间尺度参数的收敛速度。这些结果能够加深对多尺度随机系统演化行为的认识，为多尺度复杂系统的建模、仿真、参数估计、最优控制等问题提供严格的数学基础。