Corrector Theory in Random Homogenization of Partial Differential Equations

Wenjia Jing

We derive systematically a theory for the correctors in random homogenization of partial differential equations with highly oscillatory coefficients, which arise naturally in many areas of natural sciences and engineering. This corrector theory is of great practical importance in many applications when estimating the random fluctuations in the solution is as important as finding its homogenization limit. This thesis consists of three parts. In the first part, we study some properties of random fields that are useful to control corrector in homogenization of PDE. These random fields mostly have parameters in multi-dimensional Euclidean spaces. In the second part, we derive a corrector theory systematically that works in general for linear partial differential equations, with random coefficients appearing in their zero-order, i.e., non-differential, terms. The derivation is a...