  We present a general black box theorem that ensures convergence of a sequence
of stationary Markov processes, provided a few assumptions are satisfied. This
theorem relies on a control of the resolvents of the sequence of Markov
processes, and on a suitable characterization of the resolvents of the limit.
One major advantage of this approach is that it circumvents the use of the
Boltzmann-Gibbs principle: in particular, we deduce in a rather simple way that
the stationary fluctuations of the one-dimensional zero-range process converge
to the stochastic heat equation. It also allows to establish results that were
probably out of reach of existing methods: using the black box result, we are
able to prove that the stationary fluctuations of a discrete model of ordered
interfaces, that was considered previously in the statistical physics
literature, converge to a system of reflected stochastic PDEs.
