Scaling Limits of Random Walks, Harmonic Profiles, and Stationary Non-Equilibrium States in Lipschitz Domains

TL;DR

This paper investigates the scaling limits of symmetric exclusion and inclusion processes in Lipschitz domains, establishing convergence to Brownian motion with various boundary conditions and deriving hydrodynamic limits and stationary fluctuations.

Contribution

It provides the first rigorous derivation of hydrodynamic and hydrostatic limits for SEP/SIP in Lipschitz domains, including boundary condition effects, using duality techniques.

Findings

Convergence to Brownian motion with Neumann, Dirichlet, or Robin boundary conditions.

Discrete harmonic profiles converge to continuum harmonic functions.

Hydrodynamic and hydrostatic limits are rigorously established.

Abstract

We consider the open symmetric exclusion (SEP) and inclusion (SIP) processes on a bounded Lipschitz domain $\Omega$, with both fast and slow boundary. For the random walks on $\Omega$ dual to SEP/SIP we establish: a functional-CLT-type convergence to the Brownian motion on $\Omega$ with either Neumann (slow boundary), Dirichlet (fast boundary), or Robin (at criticality) boundary conditions; the discrete-to-continuum convergence of the corresponding harmonic profiles. As a consequence, we rigorously derive the hydrodynamic and hydrostatic limits for SEP/SIP on $\Omega$, and analyze their stationary non-equilibrium fluctuations. All scaling limit results for SEP/SIP concern finite-dimensional distribution convergence only, as our duality techniques do not require to establish tightness for the fields associated to the particle systems.