On hydrodynamic limits in Sinai-type random environments

TL;DR

This paper studies the large-scale behavior of interacting random walks in a Sinai-type environment, deriving a quasilinear SPDE as the hydrodynamic limit and exploring connections to Brox diffusion.

Contribution

It introduces a hydrodynamic limit for zero-range interacting particles in Sinai-type environments, resulting in a novel quasilinear SPDE with a rough drift term.

Findings

Derivation of a quasilinear SPDE as the hydrodynamic limit

Identification of a connection between particle system limits and Brox diffusion

Formal two-scale limit analysis linking microscopic and macroscopic behaviors

Abstract

We investigate the hydrodynamical behavior of a system of random walks with zero-range interactions moving in a common `Sinai-type' random environment on a one dimensional torus. The hydrodynamic equation found is a quasilinear SPDE with a `rough' random drift term coming from a scaling of the random environment and a homogenization of the particle interaction. Part of the motivation for this work is to understand how the space-time limit of the particle mass relates to that of the known single particle Brox diffusion limit. In this respect, given the hydrodynamic limit shown, we describe formal connections through a two scale limit.