Homogenization of Elliptic and Parabolic Partial Differential Equations

随机均匀化理论确定了受微观随机效应影响的现象的平均宏观行为。例如，人们可能感兴趣的是确定具有随机分布杂质的多孔材料的一般性质，或预测具有随机障碍的非均质介质中种群的演化。这种现象通常是用带有随机系数的偏微分方程（PDEs）来模拟的，这些随机系数依赖于描述非均质性的微观长度尺度。随机系数考虑了物理环境的所有可能实现，通过施加某些假设，人们可以期望，在渐近的平均情况下，几乎所有这些解都表现出相同的有效行为。

The theory of stochastic homogenization identifies the average, macroscopic behavior of a phenomenon which is subject to microscopic, random effects. For example, one may be interested in determining the general properties of a porous material with randomly distributed impurities, or predicting the evolution of a population in a heterogeneous medium with random obstacles. Such phenomena are typically modeled by partial differential equations (PDEs) with random coefficients which depend on microscopic lengthscales describing the heterogeneities. The random coefficients take into account all possible realizations of a physical environment, and by imposing certain hypotheses, one may expect that asymptotically on average, almost all such solutions exhibit the same effective behavior. The main goal of this proposal is to further the general understanding of elliptic and parabolic PDEs through the rich source of problems based in stochastic homogenization. The study of homogenization offers contributions to both theoretical and applied mathematics: homogenization often exposes many interesting problems in the analysis of the relevant equations, and it can be directly used to model physical processes. I plan to focus my efforts on two main classes of elliptic and parabolic PDEs: (a) Nondivergence Form Equations, and (b) Reaction-Diffusion Equations. Such equations serve as the primary mathematical models in stochastic control theory, finance, and geometry and chemical kinetics, combustion, and biology respectively. The projects I am proposing are motivated by the following two objectives: (1) To broaden the class of PDEs for which homogenization takes place and (2) To obtain more specific information about the process of homogenization, such as error estimates and properties of the effective behavior. The study of homogenization combines tools from several different areas of mathematics, including analysis, PDEs, dynamical systems, and probability. I am committed to using collaborative approaches for the proposed research program; drawing inspiration and techniques from various subfields. This flexible perspective promotes a unified understanding of the physical phenomena, as well as enhancing the theory for both nondivergence form and reaction-diffusion equations. Furthermore, progress in this specific research program may influence developments in the above related areas of mathematics. Aside from the immediate applications to other subfields of mathematics, the study of multiscale problems has been a source of interest for specialists in several outside areas including materials science, chemical engineering, and biology. Consequently, this work contributes towards strengthening the relationship between the mathematical theory of PDEs and applications to other scientific disciplines.