Homogenization of generalized second-order elliptic difference operators

TL;DR

This paper establishes homogenization results for generalized second-order elliptic difference operators, demonstrating convergence of discrete solutions to continuous limits under certain conditions, with applications in probability theory.

Contribution

It introduces a homogenization framework for generalized second-order difference operators in both deterministic and random environments, extending previous theories.

Findings

Homogenization holds under weak convergence and ellipticity assumptions.

Discrete models with random environments homogenize to deterministic limits.

Application to hydrodynamic limits in probability theory.

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.