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We study space-time fluctuations of a hard sphere system at thermalequilibrium, and prove that the covariance converges to the solution of alinearized Boltzmann equation in the low density limit, globally in time. Thisresult has been obtained previously in [7], by using uniform bounds on thenumber of recollisions of dispersing systems of hard spheres (as provided forinstance in [9]). We present a self-contained proof with substantialdifferences, which does not use this geometric result. This can be regarded asthe first step of a program aiming to derive the fluctuation theory of therarefied gas, for interaction potentials different from hard spheres.

Long time validity of the linearized Boltzmann equation for hard spheres: a proof without billiard theory