  We provide a rigorous derivation of the brownian motion as the limit of a
deterministic system of hard-spheres as the number of particles $N$ goes to
infinity and their diameter $\varepsilon$ simultaneously goes to $0$, in the
fast relaxation limit $\alpha = N\varepsilon^{d-1}\to \infty $ (with a suitable
diffusive scaling of the observation time). As suggested by Hilbert in his
sixth problem, we rely on a kinetic formulation as an intermediate level of
description between the microscopic and the fluid descriptions: we use indeed
the linear Boltzmann equation to describe one tagged particle in a gas close to
global equilibrium. Our proof is based on the fundamental ideas of Lanford. The
main novelty here is the detailed study of the branching process, leading to
explicit estimates on pathological collision trees.
