Density fluctuations for exclusion processes with long jumps

TL;DR

This paper investigates the stationary density fluctuations in exclusion processes with long-range jumps, showing they are described by a fractional Ornstein-Uhlenbeck process for certain parameters and by a fractional Burgers equation at a critical point.

Contribution

It characterizes the limiting behavior of density fluctuations in long-jump exclusion processes, connecting them to fractional stochastic PDEs and extending previous finite-volume results.

Findings

Density fluctuations are fractional Ornstein-Uhlenbeck processes for alpha in (0, 3/2).

At alpha = 3/2, fluctuations are tight and limit points solve the fractional Burgers equation.

Provides a rigorous link between long-range exclusion processes and fractional stochastic PDEs.

Abstract

We show that the stationary density fluctuations of exclusion processes with long jumps, whose rates are of the form $c^\pm |y-x|^{-(1+\alpha)}$ where $c\pm$ depends on the sign of $y-x$, are given by a fractional Ornstein-Uhlenbeck process for $\alpha \in (0,\frac{3}{2})$. When $\alpha =\frac{3}{2}$ we show that the density fluctuations are tight, in a suitable topology, and that any limit point is an energy solution of the fractional Burgers equation, previously introduced in \cite{GubJar} in the finite volume setting.