Hydrodynamics for the partial exclusion process in random environment

TL;DR

This paper develops a hydrodynamic limit for a partial exclusion process in a random environment with varying site occupancies, utilizing homogenization techniques and invariance principles.

Contribution

It introduces a quenched hydrodynamic limit for the partial exclusion process in a random environment, extending existing methods with new homogenization results.

Findings

Established a quenched invariance principle for a single particle in the environment.

Derived a hydrodynamic limit for the partial exclusion process.

Connected homogenization results to the particle system via self-duality.

Abstract

In this paper, we introduce a random environment for the exclusion process in $\mathbb{Z}^d$ obtained by assigning a maximal occupancy to each site. This maximal occupancy is allowed to randomly vary among sites, and partial exclusion occurs. Under the assumption of ergodicity under translation and uniform ellipticity of the environment, we derive a quenched hydrodynamic limit in path space by strengthening the mild solution approach initiated in [39] and [21]. To this purpose, we prove, employing the technology developed for the random conductance model, a homogenization result in the form of an arbitrary starting point quenched invariance principle for a single particle in the same environment, which is a result of independent interest. The self-duality property of the partial exclusion process allows us to transfer this homogenization result to the particle system and, then, apply the tightness criterion in [40].