Boundary driven Markov gas: duality and scaling limits

TL;DR

This paper develops a unified framework for boundary driven particle systems, demonstrating duality properties and showing how the boundary driven Brownian gas emerges as a diffusive scaling limit of random walks with reservoirs.

Contribution

It introduces a general duality framework for boundary driven systems in discrete and continuum settings, and establishes the Brownian gas as a scaling limit.

Findings

Duality with an absorbed boundary process is established.

The invariant measure is a Poisson point process solving a Dirichlet problem.

The boundary driven Brownian gas is derived as a diffusive scaling limit.

Abstract

Inspired by the recent work of Bertini and Posta, who introduced the boundary driven Brownian gas on $[0,1]$, we study boundary driven systems of independent particles in a general setting, including particles jumping on finite graphs and diffusion processes on bounded domains in $\mathbb{R}^d$. We prove duality with a dual process that is absorbed at the boundaries, thereby creating a general framework that unifies dualities for boundary driven systems in the discrete and continuum setting. We use duality first to show that from any initial condition the systems evolve to the unique invariant measure, which is a Poisson point process with intensity the solution of a Dirichlet problem. Second, we show how the boundary driven Brownian gas arises as the diffusive scaling limit of a system of independent random walks coupled to reservoirs with properly rescaled intensity.