Homogenization of Elliptic and Parabolic Partial Differential Equations

均质化的数学理论确定了受微观影响的现象的平均宏观行为。例如，人们可能对确定多孔材料的一般性质或预测物质在非均质介质中的演变感兴趣。这种现象通常用偏微分方程来模拟，偏微分方程依赖于描述非均质性的微观长度尺度。均匀化是用更光滑的宏观模型逼近这些详细方程的过程。首席研究员将集中研究所谓的随机均质化问题，即微观效应是随机分布的。这样的模型对于建立一个健壮的框架来表示受不确定性影响的“典型”物理环境具有重要意义。一般来说，均匀化的研究结合了几个不同数学领域的工具，包括分析、偏微分方程、动力系统和概率论。首席研究员承诺在项目中采用合作的方式。这种灵活的视角促进了对物理现象的统一理解，并加强了相关方程的理论。首席研究员将集中精力研究两类主要的椭圆型和抛物型偏微分方程：(a)非发散型方程，它描述一般的扩散过程，经常用于随机控制理论和几何的研究；(b)反应-扩散方程，其解代表前沿进化，是化学动力学、燃烧和生物学的主要数学模型。该研究将包括由以下两个目标驱动的各种子项目的建议：(1)表明均匀化适用于比先前预期的更广泛的偏微分方程类别；(2)获得比目前已知的关于均质化过程的更具体的信息，例如误差估计或有效行为的性质。最初在随机均匀化中提出的问题通常在概率论中具有等效的公式。因此，该研究可能会导致随机环境下的随机行走、第一通道渗透和大偏差原理的研究取得进展。

The mathematical theory of homogenization identifies the average, macroscopic behavior of a phenomenon that is subject to microscopic effects. For example, one may be interested in determining the general properties of a porous material, or predicting the evolution of a substance traveling through a heterogeneous medium. Such phenomena are typically modeled by partial differential equations that depend on microscopic length-scales describing the heterogeneities. Homogenization is the process of approximating such detailed equations with smoother, macroscopic models. The principal investigator will focus on the subject of so-called stochastic homogenization, in which the microscopic effects are randomly distributed. Such models are significant for developing a robust framework to represent "typical" physical settings that are subject to uncertainty. Generally speaking, the study of homogenization combines tools from several different areas of mathematics, including analysis, partial differential equations, dynamical systems, and probability theory. The principal investigator is committed to using collaborative approaches to the project. This flexible perspective promotes a unified understanding of the physical phenomena, as well as enhancing the theory of the relevant equations. The principal investigator will focus her efforts on two main classes of elliptic and parabolic partial differential equations: (a) non-divergence-form equations, which describe general diffusion processes and are frequently used in the study of stochastic control theory and geometry; and (b) reaction-diffusion equations, solutions of which represent front-like evolution and serve as the primary mathematical models in chemical kinetics, combustion, and biology. The research will encompass proposes a variety of sub-projects that are motivated by the following two objectives: (1) to show that homogenization is applicable to a broader class of partial differential equations than previously expected; and (2) to obtain more specific information about the process of homogenization than is currently known, such as error estimates or properties of the effective behavior. Questions posed initially in stochastic homogenization typically have equivalent formulations in probability theory. Consequently, the research may lead to progress in the study of random walks in random environments, first passage percolation, and large deviation principles.