随机偏微分方程的有效宏观建模和稀有事件

This project develops effective mathematical methodology to extract explicit and accurate macroscopic models from complex stochastic physical and engineering systems. The important class of systems we address are those microscopically described in space-time by stochastic partial differential equations(SPDEs). Due to micro-macro and nonlinear interaction, randomness on the microscopic scale feeds into macroscopic dynamics and needs to be accounted for as in our planned modelling. Thermally activated transitions from one stable region to another happen with a small probability that we also will quantify as rare events. These results will provide effective theory and methodologies for the simulation and understanding of large, noisy complex systems.In a little more detail, this project aims to contribute the following: 1. Derive explicit and accurate, effective macroscopic models of complex systems described by SPDEs which have multiple metastable states. The outcome will be much greater understanding of key issues in stochastic modelling and of the relationship between macroscale and microscale stochastic dynamics. 2. Develop random dynamical theory methodology for macroscale models suitable for computational simulation. Then other researchers will be empowered to generate accurate and efficient macroscale stochastic simulations for their microscale SPDEs without resolving unnecessary details. 3. Develop a practical theory of large deviations estimates for multiscale SPDEs to empower prediction of any potential atypical behaviour and describe the mechanism of rare events which governs the metastability of complex system with stochastic forcing.This will greatly increase understanding of risk in complex stochastic systems. The results of the project will provide effective theory and method to simulate and understand complex system forced by noise.

本项目目标是发展有效的数学方法，从描述物理和工程中复杂现象的细致模型中建立一个有效宏观模型。这些细致模型是由一类重要的时空微观尺度上的随机偏微分方程来描述。由于宏观- - 微观以及非线性之间的相互作用，微观尺度上的随机因素会体现在宏观动力行为上，从而宏观上建立的模型是一个随机模型。另外随机激励使得系统中稳定区域之间的小概率的迁移发生，这称为稀有事件。具体地，本项目主要研究： 1. 从由随机偏微分方程描述的具有多个稳定态的复杂系统中得到精确的有效地宏观模型。 2. 发展随机动力系统方法，对随机偏微分方程建立有效的数值离散模型。 3. 对具有多尺度的随机偏微分方程建立大偏差理论，并刻划复杂系统中稀有事件的发生机制。 本项目的结果将对模拟和理解噪声影响的复杂系统提供有效的理论和方法。

物理和工程中复杂现象的细致模型中通常具有涨落，而且该涨落与宏观时间尺度具有很大的分离，因此我们需要建立一个有效宏观模型。本项目在理论上对几类具有快速涨落的随机偏微分方程建立有效宏观模型，分析了系统中非线性，边界涨落，小扰动与随机力相互作用对模型的影响。具体得到如下几个结果：.1. 利用扩散逼近方法研究了具有随机对流的Burgers方程中的随机自相似性。.2. 利用扩散逼近方法对具有边界快速涨落的偏微分方程建立有效逼近模型。.3. 对具有奇异扰动的随机Klein-Gordon-Schrodinger方程建立逼近模型并给出大偏差估计。.4. 对具有奇异扰动的随机波动方程建立了其随机惯性流形的逼近。.5. 对具有强相互作用的随机波动方程建立空间齐次逼近模型。.6. 具有时空白噪声驱使的分数Burgers方程Kolmogorov算子的m-耗散性。.这些结果对随机偏微分方程的宏观约化提供很好的理论支持，对随机偏微分方程理论是一个重要补充。