  We investigate the fluctuations around the average density profile in the
weakly asymmetric exclusion process with open boundaries in the steady state.
We show that these fluctuations are given, in the macroscopic limit, by a
centered Gaussian field and we compute explicitly its covariance function. We
use two approaches. The first method is dynamical and based on fluctuations
around the hydrodynamic limit. We prove that the density fluctuations evolve
macroscopically according to an autonomous stochastic equation, and we search
for the stationary distribution of this evolution. The second approach, which
is based on a representation of the steady state as a sum over paths, allows
one to write the density fluctuations in the steady state as a sum over two
independent processes, one of which is the derivative of a Brownian motion, the
other one being related to a random path in a potential.
