一类多尺度非局部随机系统的平均化原理

Multiscale models are attracting great attention in many disciplines such as mathematics, physics, biology and engineering. One of the central problems in multiscale models is to reduce the model, the commonly used reduction methods are averaging principle, slow manifolds and other methods. We have now established an averaging results for class of non-Lipstitz multi-valued stochastic differential equations. On this basis, we consider the averaging problem of nonlocal coupled stochastic partial differential equations with fast and slow time scales. It can be seen that large-scale effects, random effects and non-local operators are the main difficulties in solving this problem. In the framework of stochastic dynamical systems and stochastic partial differential equations, we will study the following questions: the necessary conditions so as that there exist a reduction equation which approximates the dominant component of the nonlocal stochastic system with two time scales, the numerical scheme for the reduction equation, 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 nonlocal stochastic systems with multiple scales. It also provides a rigorous theoretical basis for modeling, simulation, parameter estimation, optimal control for nonlocal 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，常用的约化方法有平均化、慢流形等方法。目前我们已建立了一类非李普希兹条件下多值随机微分方程的平均化结果，在此基础上我们考虑具有快慢两个时间尺度非局部耦合随机偏微分方程的平均化问题。可以看到，大尺度效应、随机影响以及非局部算子是解决该问题的几个主要难点。在随机动力系统和随机偏微分方程理论的框架下，我们将研究如下问题：约化方程存在性以及约化方程逼近原系统所需的必要条件、约化方程的数值算法、主要分量与约化方程的解过程在强收敛(轨道的逼近)及弱收敛(分布的逼近)意义下关于时间尺度参数的收敛速度。这些结果能够加深对非局部多尺度随机系统演化行为的认识， 为非局部多尺度复杂随机系统的建模、 仿真、 参数估计、最优控制等问题提供严格的数学基础。

非局部问题已经逐步渗透到了众多科学和工程领域。近年来，非局部微积分在粘弹性力学、电化学、生物学、生物工程、控制理论等领域得到了日益广泛的应用。 尽管其中仍有许多数学理论悬而未决，但由于非局部微积分算子具有独特的记忆性和遗传性，其较经典的微积分算子有着更为显著的优势，因而其理论和应用方面的研究受到了广泛的关注。 本项目主要关注于非局部双尺度随机微分方程的约化问题，主要通过平均化的手段攫取非局部双尺度随机微分方程的有效简单模型，从而用来替代原来较复杂的系统，进而在理论分析以及数值分析方面提供强有力的理论基础。目前项目正在有序进行，初步得到了一系列相关结果，这些结果能够加深对非局部多尺度随机系统演化行为的认识，为非局部多尺度复杂随机系统的建模、 仿真、 参数估计、最优控制等问题提供严格的数学基础。