Homogenization of Elliptic and Parabolic Partial Differential Equations

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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.