Symmetric exclusion as a random environment: hydrodynamic limits

TL;DR

This paper studies a random walk in a dynamic environment created by a symmetric exclusion process, proving a hydrodynamic limit and deriving an ODE for the walk's macroscopic behavior.

Contribution

It establishes a hydrodynamic limit for the exclusion process viewed from the walk without relying on explicit invariant measures, extending to variants of the model.

Findings

Hydrodynamic limit theorem for the exclusion process as seen from the walk

Derivation of an ODE describing the walk's macroscopic evolution

Development of a replacement lemma without explicit invariant measures

Abstract

We consider a one-dimensional continuous time random walk with transition rates depending on an underlying autonomous simple symmetric exclusion process starting out of equilibrium. This model represents an example of a random walk in a slowly non-uniform mixing dynamic random environment. Under a proper space-time rescaling in which the exclusion is speeded up compared to the random walk, we prove a hydrodynamic limit theorem for the exclusion as seen by this walk and we derive an ODE describing the macroscopic evolution of the walk. The main difficulty is the proof of a replacement lemma for the exclusion as seen from the walk without explicit knowledge of its invariant measures. We further discuss how to obtain similar results for several variants of this model.