Quantitative Stochastic Homogenization and Renormalization Methods

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This research project is focused on the development of new mathematical ideas and tools for understanding the macroscopic properties of physical systems with many small-scale irregularities. Composite materials, for example, are manufactured by intermingling two or more constituent materials and often have very different physical properties (heat or electrical conduction, for example) from the individual components. Surprisingly, the properties of the composite material are typically not a simple average of those of its components--it matters greatly how the (microscopic) intermingling is arranged. We want to understand precisely how the large number of microscopic interactions give rise to the macroscopic behavior of such systems, to be able to predict the macroscopic properties of the system by carefully looking at the microscopic arrangement of a sample, and to know exactly how large such a sample must be before our predictions are accurate. We proceed by developing and analyzing idealized mathematical models that display the key features and complexity of real physical systems, but still simple enough that they can be studied mathematically. Very often, our models are partial differential equations with stochastic coefficients--that is, the underlying microscopic behavior of the system is assumed to be random--and the goal is to understand the statistics of the system. Our objective is to develop mathematical approaches that are (i) informed by the physics of the systems, (ii) robust--they can help us solve a variety of other, similar problems arising in, for example, statistical physics and probability theory, and (iii) quantitative--they lead to quantitative estimates of the uncertainty and insight into the development of numerical algorithms for simulation and prediction. This project provides research training opportunities for graduate students.Obtaining an "averaged" partial differential equation that describes the large-scale behavior of an underlying highly heterogeneous equation with many degrees of freedom (for instance, one with random coefficients) is referred to as "homogenization" in the mathematical literature. The homogenized equation is typically much simpler than the "true" heterogeneous one, and hence easier to work with. It is therefore important to understand precisely how well the homogenized equation approximates the true equation, and this is the goal of homogenization theory. Mathematicians have recently developed a complete quantitative theory of stochastic homogenization for elliptic equations, and this theory has already been shown to have some surprising and important applications to probability theory and statistical mechanics. The present project aims to continue in this direction, by applying and developing homogenization techniques for gradient lattice models and related models in statistical physics, to the study of hypoelliptic diffusions arising in kinetic theory (modeling gases and plasmas) and toward understanding the behavior of diffusions forced by random vector fields.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.