  We consider the weakly asymmetric simple exclusion process on a ring, driven
out of equilibrium by tilting the dynamics so as to enforce a macroscopic
current of particles on a large time interval. In this current-biased dynamics,
the tilt by the current makes the dynamics non-local, non homogeneous and
induces long-range correlations. We compute the correlation structure in the
large time, large size limit for a certain range of asymmetry and current
strength, recovering heuristic results of Bodineau et al. arXiv:0807.2394. In
addition, in this range of parameters, we characterise the full dynamics of
fluctuations around the optimal density profile in the current-biased dynamics.
The key ingredient at the microscopic scale is a precise relative entropy
estimate at the level of correlations. We also discuss how to remove the
(technical) restriction on the range of parameters.
