Stochastic Homogenization of Parabolic Equations with Lower-order Terms

Abstract

The study of homogenization results has long been a central focus in the field of mathematical analysis, particularly for equations without lower-order terms. However, the importance of studying homogenization results for parabolic equations with lower-order terms cannot be understated. In this study, we aim to extend the analysis to homogenization for the general parabolic equation with random coefficients: \begin{equation*} \partial_{t}p^ε-\nabla\cdot\left(\mathbf{a}\left( \dfrac{x}ε,\dfrac{t}{ε^2}\right)\nabla p^ε\right)-\mathbf{b}\left( \dfrac{x}ε,\dfrac{t}{ε^2}\right)\nabla p^ε-\mathbf{d}\left( \dfrac{x}ε,\dfrac{t}{ε^2}\right) p^ε=0. \end{equation*} Moreover, we establish the Caccioppoli inequality and Meyers estimate for the generalized parabolic equation. By using the generalized Meyers estimate, we get the weak convergence of $p^{\hat{I}}\mu$ in $H^1$.