Branching processes, trees and the Boltzmann equation

Using the formalism of random trees, we construct a process solution of the spacehomogeneous Boltzmann equation. We deduce a simulation method having relationships with both the Nanbu's method and with Bird's method. The efficiency is clear for the Kac's caricature and some scalar Boltzmann cases because the algorithm is then explicit. This construction is also the tool for proving a geometric convergence to the equilibrium.