  We investigate the fluctuations of cumulative density of particles in the
asymmetric simple exclusion process with respect to the stationary distribution
(also known as the steady state), as a stochastic process indexed by $[0,1]$.
In three phases of the model and their boundaries within the fan region, we
establish a complete picture of the scaling limits of the fluctuations of the
density as the number of sites goes to infinity. In the maximal current phase,
the limit fluctuation is the sum of two independent processes, a Brownian
motion and a Brownian excursion. This extends an earlier result by Derrida et
al. (2004) for totally asymmetric simple exclusion process in the same phase.
In the low/high density phases, the limit fluctuations are Brownian motion.
Most interestingly, at the boundary of the maximal current phase, the limit
fluctuation is the sum of two independent processes, a Brownian motion and a
Brownian meander (or a time-reversal of the latter, depending on the side of
the boundary). Our proofs rely on a representation of the joint generating
function of the asymmetric simple exclusion process with respect to the
stationary distribution in terms of joint moments of a Markov processes, which
is constructed from orthogonality measures of the Askey--Wilson polynomials.
