We study the behavior of the nonlinear Markov process associated to the Boltzmann equation under both hyperbolic and parabolic space-time scalings. In the first case the limit of the process is the solution of an o.d.e, with vector field given by a solution of the Euler equation, while in the second case the limit of the process, in the incompressible case, turns out to be a diffusion process whose drift is a solution of the incompressible Navier-Stokes equation.