Equilibrium fluctuations for the weakly asymmetric discrete Atlas model

TL;DR

This paper studies the equilibrium fluctuations of a weakly asymmetric discrete Atlas model, showing that its macroscopic limit follows a stochastic heat equation with boundary conditions, extending previous work in the field.

Contribution

It generalizes the discrete Atlas model to include weak asymmetry and derives the associated stochastic PDE with boundary conditions, expanding understanding of boundary effects in such models.

Findings

The fluctuation field converges to a solution of a stochastic heat equation.

Boundary conditions depend on model parameters, being either Neumann or Robin.

The work extends previous models by incorporating weak asymmetry and boundary effects.

Abstract

This contribution aims at presenting and generalizing a recent work of Hernandez, Jara and Valentim [DOI:10.1016/j.spa.2016.06.026]. We consider the weakly asymmetric version of the so-called discrete Atlas model, which has been introduced there. Precisely, we look at some equilibrium fluctuation field of a weakly asymmetric zero-range process which evolves on a discrete half-line, with a source of particles at the origin. We prove that its macroscopic evolution is governed by a stochastic heat equation with Neumann or Robin boundary conditions, depending on the range of the parameters of the model.