Diffusive fluctuations of long-range symmetric exclusion with a slow barrier

TL;DR

This paper studies the equilibrium fluctuations of a long-range symmetric exclusion process on the integers, revealing how slow barriers and jump rates influence the resulting stochastic partial differential equations under diffusive scaling.

Contribution

It characterizes the limiting SPDEs for long-range exclusion processes with slow barriers, extending understanding of boundary effects in long-range interacting particle systems.

Findings

Different boundary conditions (Neumann, Robin, none) depending on parameters

Explicit connection between jump rate decay and resulting SPDEs

Extension of fluctuation results to long-range interactions with barriers

Abstract

In this article we obtain the equilibrium fluctuations of a symmetric exclusion process in $\mathbb{Z}$ with long jumps. The transition probability of the jump from $x$ to $y$ is proportional to $|x-y|^{-\gamma-1}$. Here we restrict to the choice $\gamma \geq 2$ so that the system has a diffusive behavior. Moreover, when particles move between $\mathbb{Z}_{-}^{*}$ and $\mathbb N$, the jump rates are slowed down by a factor $\alpha n^{-\beta}$, where $\alpha>0$, $\beta\geq 0$ and $n$ is the scaling parameter. Depending on the values of $\beta$ and $\gamma$, we obtain several stochastic partial differential equations, corresponding to a heat equation without boundary conditions, or with Robin boundary conditions or Neumann boundary conditions.