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

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 − α if x > 0 and = c − | x | − 1 − α if x < 0 . Above, c ± ≥ 0 , α > 0 are parameters. We prove that for 0 < α < 3 / 2 , the density fluctuation fields converge to the stationary solution of a coupled fractional Ornstein–Uhlenbeck process, and for α = 3 / 2 , the limit points are concentrated on stationary energy solutions of a coupled fractional Burgers equation.