  We develop a rigorous theory of hard-sphere dynamics in the kinetic regime,
away from thermal equilibrium. In the low density limit, the empirical density
obeys a law of large numbers and the dynamics is governed by the Boltzmann
equation. Deviations from this behaviour are described by dynamical
correlations, which can be fully characterized for short times. This provides
both a fluctuating Boltzmann equation and large deviation asymptotics.
