多尺度随机微分方程的平均原理

This project will focus on mathematical problems in complexity science studying the complex systems, these problems appear in mathematics, physics, chemistry, finance, and earth science, life science, information science, materials science etc., and are new important problems. The key difficulty to solve these problems is to deal with multiscale random perturbations，and one of the main tools is the averaging principle. In this project, we will study strong convergence, weak convergence and convergence rate of the averaging principle of multiscale stochastic partial differential equations with infinity degrees of freedom, the averaging principle of multiscale stochastic partial differential equations driven by non-Gauss processes, moderate deviations and large deviations of the averaging principle for the multiscale stochastic partial differential equations, the averaging principle of multiscale forward-backward stochastic differential equations, the averaging principle of multiscale stochastic differential equations with non-regular coefficients, the failure conditions of averaging principle and strict mathematical form of possible alternative conditional averaging principle when averaging principle fails. These results can simplify the real complex systems for the reduction system of slow process, and can also be used in random sampling, parameter estimation, numerical integration schemes, filtering and stochastic optimal control of multiscale stochastic differential equations.

本项目将研究以复杂系统为研究对象的复杂性科学中的数学问题，这些问题是当今数学、物理学、化学、金融学、地球科学、生命科学、信息科学、材料科学等高新领域的核心前沿问题，其关键困难是要处理多尺度随机扰动问题。平均原理是解决多尺度扰动问题的主要工具之一。在本项目中，我们将重点研究具有无穷个自由度的多尺度随机偏微分方程平均原理的强收敛、弱收敛及其收敛速度，非Gauss过程驱动的多尺度随机偏微分方程的平均原理，多尺度随机偏微分方程平均原理的大偏差、中偏差原理，多尺度正倒向随机微分方程的平均原理，具有非正则系数多尺度随机微分方程的平均原理，平均原理失效的条件和此时可能作为替代的条件平均原理的严格数学形式。这些结果期望能对现实复杂系统进行降维得到慢过程的约化系统，同时也为多尺度随机微分方程的随机抽样、参数估计、数值计算、滤波、随机最优控制提供理论支持。

在项目执行期内，本项目研究了几类随机偏微分方程的平均原理，包括抛物方程与双曲方程耦合的多尺度随机偏微分方程，多尺度随机FitzHugh-Nagumo系统，带跳的多尺度随机FitzHugh-Nagumo系统，由柱Brown运动与Poisson随机测度联合驱动的多尺度随机偏微分方程，以及多值随机微分方程的平均原理，得到了平均原理强收敛和弱收敛的收敛速度，以及研究了证明收敛的工具Borel-Cantelli引理，同时研究了分数Brown运动在Holder范数下的拟必然大偏差原理，与分数Brown单的增量在Holder范数下的拟必然泛函极限定理，这些结果为后续工作打下了基础，研究结果发表在Stochastic Processes and their Applications，Journal of Mathematical Physics，Stochastic analysis and applications，Discrete and Continuous Dynamical Systems-Series B，J. Math. Anal. Appl.等杂志上。