Hydrodynamic limit of symmetric exclusion processes in inhomogeneous media

TL;DR

This paper reviews a unified approach linking the hydrodynamic limit of symmetric exclusion processes to homogenization problems, highlighting its generality and recalling solutions from previous works.

Contribution

It clarifies the relation between hydrodynamic limits and homogenization, and summarizes how the homogenization problem has been solved in prior research.

Findings

Unified approach simplifies analysis of exclusion processes

Homogenization problem has been effectively solved in previous works

Method applies to a broad class of symmetric exclusion processes

Abstract

In \cite{J} M. Jara has presented a method, reducing the proof of the hydrodynamic limit of symmetric exclusion processes to an homogenization problem, as unified approach to recent works on the field as \cite{N}, \cite{F1}, \cite{F2} and \cite{FJL}. Although not stated in \cite{J}, the reduction of the hydrodynamic limit to an homogenization problem was already obtained (in a different way) in \cite{N}, \cite{F1}. This alternative and very simple relation between the two problems goes back to an idea of K.\ Nagy \cite{N}, is stated in \cite{F1}[Section B] for exclusion processes on $\bbZ^d$ and, as stressed in \cite{F2}, is completely general. The above relation has been applied to \cite{N}, \cite{F1}, \cite{F2} and \cite{FJL} and could be applied to other symmetric exclusion processes, mentioned in \cite{J}. In this short note we briefly recall this unified approach in a complete general setting. Finally, we recall how the homogenization problem has been solved in the above previous works.