Dynamics in Random Media: from Homogenization to Stochasticity

这项研究致力于了解随机扰动对复杂系统行为的影响。随机性在物理学、应用科学和工程学中广泛存在，包括统计力学、大气科学、材料工程、海洋学等。系统中通常涉及各种时间或空间尺度，了解随机性在这些尺度上的传播并控制其总体影响对于勘探地球物理学、材料设计、气象学和天气预报等许多应用至关重要。对此类涉及随机性的多尺度动力系统进行建模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)。该项目旨在推导具有粗糙随机系数的偏微分方程的简化有效模型，并展示复杂系统中随机性传播机制的一些一般特征。拟议的研究涉及随机均匀化、随机介质中的波和随机偏微分方程。主要研究者计划研究（i）随机均质化中的随机波动； (ii) 波在随机介质中传播的有效模型； (iii) 超出均质化范围的随机偏微分方程的收敛性以及某些奇异随机偏微分方程的统计特性。该研究将涉及分析、概率和数值方法的工具。该教育计划包括对研究生和本科生进行概率、分析和应用交叉方面的培训，目标是向学生介绍研究和 STEM 相关职业发展不可或缺的分析和概率工具。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影响审查标准进行评估，被认为值得支持。

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.