CAREER: Partial Differential Equation and Randomness

这项研究的目的是对非均匀环境中的一些动力学进行数学研究，如晶体沉积、污染物在非均匀流中的扩散和海洋中的声音传播。偏微分方程组(PDE)是模拟这类现象的主要数学工具，为了解释信息的缺乏或模型的不确定性，很自然地在方程中加入随机性。这类问题的一个普遍特征是时间和空间尺度的广泛，一直延伸到观测兴趣的尺度。由于数值模拟在微观尺度上几乎是不可能的，因此对于这些问题，简化和有效地描述物理量，如表面高度、污染物密度和波强是非常必要的。该项目的目标是研究相关的随机偏微分方程组，并为简化的有效模型在不同的近似条件下奠定数学基础。该项目的有机组成部分是与研究相结合的教育项目。该教育计划包括通过设计高级课程、指导本科生研究和为早期职业研究人员组织研讨会，在概率论、分析和应用的交叉点对研究生和本科生进行持续培训。目标是向学生介绍分析和概率工具，这些工具对于学术研究和工程应用都是不可或缺的。本课题主要研究随机偏微分方程组、随机环境中的扩散、随机介质中的波传播等方面的问题。通过结合渐近展开、泛函不等式和随机分析等工具，主要研究人员将研究(I)非线性随机偏微分方程组模拟界面生长；(Ii)定向聚合物在随机环境中的行为；(Iii)随机介质中波传播的有效模型。该项目的主要主题之一是分析观测数据对随机扰动的依赖性，并了解非线性和随机性之间的相互作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This research aims at carrying out mathematical studies of some dynamics in heterogeneous environments, such as crystal deposition, spreading of pollutants in inhomogeneous flows, and sound propagation in the ocean. Partial differential equations (PDEs) serve as the principal mathematical tool to model such phenomena, and to account for lack of information or model uncertainty it is natural to add randomness to the equations. A ubiquitous feature of such problems is a wide range of temporal and spatial scales, extending all the way up to the scale of observational interest. As numerical simulations are practically impossible on the microscopic scale, a simplified and effective description of the physical quantities, such as the surface height, the density of pollutants, and the wave intensity, is extremely desirable for these problems. The goal of the project is to study the relevant random PDEs, and to lay down mathematical foundations for simplified effective models in various approximation regimes. Organic part of this project is the educational program that is integrated with research. The educational program includes a sustained training of graduate and undergraduate students at the intersection of probability theory, analysis, and applications, through designing advanced courses, supervising undergraduate research, and organizing workshops for early career researchers. The goal is to introduce students to the analytic and probabilistic tools that are indispensable for both academic research and engineering applications. The focus of this project is in the areas of stochastic PDEs, diffusion in random environment, and wave propagation in random media. By combining tools, such as asymptotic expansions, functional inequalities and stochastic analysis, the principal investigator will study (i) nonlinear stochastic PDEs modeling interface growth; (ii) behavior of directed polymers in random environment; (iii) effective models of wave propagation in random media. One of principal themes of the project is to analyze the dependence of observables on the random perturbations, and to understand the interplay between the nonlinearity and the randomness.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.