Stationary fluctuations for a multi-species zero range process with long jumps

Abstract

We consider stationary fluctuations for the multi-species zero range process with long jumps in one dimension, where the underlying transition probability kernel is $p\left(x\right)=c_{+}{\left|x\right|}^{-1-\alpha }$ if $x>0$ and $=c_{-}{\left|x\right|}^{-1-\alpha }$ if $x<0$. Above, $c_{\pm }\ge 0,\alpha >0$ are parameters. We prove that for $0<\alpha <3/2$, the density fluctuation fields converge to the stationary solution of a coupled fractional Ornstein–Uhlenbeck process, and for $\alpha =3/2$, the limit points are concentrated on stationary energy solutions of a coupled fractional Burgers equation.