Dynamics in Random Media: from Homogenization to Stochasticity

本研究致力于了解随机扰动对复杂系统行为的影响。随机性广泛存在于物理学、应用科学和工程学中，包括统计力学、大气科学、材料工程、海洋学等，系统中通常涉及不同的时间或空间尺度，了解随机性在这些尺度上的传播并控制其总体影响对于许多应用如勘探地球物理、材料设计、气象学和天气预报。建模这种涉及随机性的多尺度动力系统的主要数学工具是通过具有粗糙随机系数的偏微分方程（PDE）。本项目的目标是导出具有粗糙随机系数的偏微分方程的简化有效模型，并展示随机性在复杂系统中传播机制的一些一般特征。建议的研究领域是随机均匀化，随机介质中的波，随机偏微分方程。主要研究者计划研究（i）随机均匀化中的随机波动;（ii）随机介质中波传播的有效模型;（iii）超越均匀化制度的随机PDE的收敛性和一些奇异随机PDE的统计特性。这项研究将涉及分析、概率和数值方法等工具。教育计划包括在概率，分析和应用的交叉点的研究生和本科生的培训，目标是向学生介绍分析和概率工具，这是研究和STEM不可或缺的-该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的评估被认为是值得支持的。影响审查标准。

This research is devoted to understanding the effects of random perturbations on behavior of complex systems. Stochasticity abounds in physics, applied science and engineering, ranging from statistical mechanics, atmospheric science, materials engineering, oceanography and so on. There are typically various temporal or spatial scales involved in the system, and to understand the propagation of stochasticity through these scales and to control its overall impact is of crucial importance for many applications such as exploration geophysics, materials design, meteorology and weather forecasting. The main mathematical tool to model such multi-scale dynamical systems involving stochasticity is through partial differential equations (PDE) with rough random coefficients. The project aims at deriving simplified effective models of those PDE with rough random coefficients and exhibiting some general features of the mechanism of stochasticity propagation through complicated systems. The proposed research is in the areas of stochastic homogenization, wave in random media, and stochastic PDE. The principle investigator plans to study (i) the random fluctuations in stochastic homogenization; (ii) the effective models of wave propagation in random media; (iii) the convergence to stochastic PDE beyond the homogenization regime and the statistical properties of some singular stochastic PDE. The study will involve tools from analysis, probability and numerical methods. The educational program includes the training of graduate and undergraduate students at the intersection of probability, analysis, and applications, and the goal is to introduce students the analytic and probabilistic tools which are indispensable for both the research and the STEM-related career development.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.