  For finite size Markov chains, the Donsker-Varadhan theory fully describes
the large deviations of the time averaged empirical measure. We are interested
in the extension of the Donsker-Varadhan theory for a large size
non-equilibrium system: the one-dimensional symmetric simple exclusion process
connected with reservoirs at different densities. The Donsker-Varadhan
functional encodes a variety of scales depending on the observable of interest.
In this paper, we focus on the time-averaged two point correlations and
investigate the large deviations from the steady state behaviour. To control
two point correlations out of equilibrium, the key input is the construction of
a simple approximation to the invariant measure. This approximation is
quantitative in time and space as estimated through relative entropy bounds
building on the work of Jara and Menezes arXiv:1810.09526.
