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 stochastic systems. It can be seen that large-scale effects、random effects、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非局部随机系统的平均化问题。可以看到大尺度效应、随机影响、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建模、仿真、参数估计、最优控制等问题提供严格的数学基础。

这是一个访问学者项目，根据这个项目设立的目的，我们与相关访问学者进行了如下问题的讨论并取得了结果：.1）.为了研究非局部快-慢随机系统中快变量系统的遍历性质和慢变量系统的平均化原理，我们着重研究了一个更基本的问题，最终连续马氏半群的遍历性质，证明了弱*平均收敛等性质。.2）.我们研究了一类分数(非局部)随机微分方程的适定性和Caratheodory 逼近问题以及一类随机微分方程的平均化原理。.3）.我们研究了一次加强随机游动的长时间行为，得到了该过程在一层梯子图上的常返性和有限树上经验测度的大偏差原理。