  It has been known since Lanford [19] that the dynamics of a hard sphere gas
is described in the low density limit by the Boltzmann equation, at least for
short times. The classical strategy of proof fails for longer times, even close
to equilibrium. In this paper, we introduce a duality method coupled with a
pruning argument to prove that the covariance of the fluctuations around
equilibrium is governed by the linearized Boltzmann equation globally in time
(including in diffusive regimes). This method is much more robust and simple
than the one devised in [4] which was specific to the 2D case.
