  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.
