Stationary fluctuations for the WASEP with long jumps and infinitely extended reservoirs

TL;DR

This paper investigates the stationary fluctuations of a weakly asymmetric exclusion process with long jumps and infinite reservoirs, showing they are described by either the Ornstein-Uhlenbeck process or the stochastic Burgers equation depending on asymmetry strength.

Contribution

It establishes the limiting behavior of stationary fluctuations for a WASEP with long jumps and infinite reservoirs, connecting to well-known stochastic PDEs.

Findings

Fluctuations follow the Ornstein-Uhlenbeck process under weak asymmetry.

Fluctuations follow the stochastic Burgers equation under stronger asymmetry.

Results depend on the asymmetry parameter controlling the dynamics.

Abstract

We study a weakly asymmetric exclusion process with long jumps and with infinitely many extended reservoirs. We prove that the stationary fluctuations of the process are governed by the generalized Ornstein-Uhlenbeck process or the stochastic Burgers equation with Dirichlet boundary conditions depending on the strength of the asymmetry of the dynamics.