Homogenization of generalized second-order elliptic difference operators

Abstract

Fix a function $W(x_1,\ldots,x_d) = \sum_{k=1}^d W_k(x_k)$ where each $W_k: \mathbb{R} \to \mathbb{R}$ is a strictly increasing right continuous function with left limits. For a diagonal matrix function $A$, let $\nabla A \nabla_W = \sum_{k=1}^d \partial_{x_k}(a_k\partial_{W_k})$ be a generalized second-order differential operator. We are interested in studying the homogenization of generalized second-order difference operators, that is, we are interested in the convergence of the solution of the equation $$\lambda u_N - \nabla^N A^N \nabla_W^N u_N = f^N$$ to the solution of the equation $$\lambda u - \nabla A \nabla_W u = f,$$ where the superscript $N$ stands for some sort of discretization. In the continuous case we study the problem in the context of $W$-Sobolev spaces, whereas in the discrete case the theory is developed here. The main result is a homogenization result. Under minor assumptions regarding weak convergence and ellipticity of these matrices $A^N$, we show that every such sequence admits a homogenization. We provide two examples of matrix functions verifying these assumptions: The first one consists to fix a matrix function $A$ with some minor regularity, and take $A^N$ to be a convenient discretization. The second one consists on the case where $A^N$ represents a random environment associated to an ergodic group, which we then show that the homogenized matrix $A$ does not depend on the realization $\omega$ of the environment. Finally, we apply this result in probability theory. More precisely, we prove a hydrodynamic limit result for some gradient processes.