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 $p(x)=c_{+}|x{|}^{-1-\alpha }$ if $x>0$ and $=c_{-}|x{|}^{-1-\alpha }$ 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 to a coupled fractional Burgers equation.

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