Dynamics in Random Media: from Homogenization to Stochasticity

本研究致力于理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、应用科学和工程领域，包括统计力学、大气科学、材料工程、海洋学等。系统通常涉及各种时间或空间尺度，通过这些尺度了解随机性的传播并控制其总体影响对于勘探地球物理，材料设计，气象和天气预报等许多应用至关重要。对这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带有粗糙随机系数的偏微分方程。本课题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，并揭示复杂系统中随机性传播机制的一些一般特征。本文拟在随机均匀化、随机介质中的波和随机偏微分方程等方面进行研究。主要研究者计划研究：(1)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；(3)超越均匀区向随机偏微分方程的收敛性和一些奇异随机偏微分方程的统计性质。该研究将涉及分析、概率和数值方法等工具。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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.