  We consider the motion of a particle governed by a weakly random Hamiltonian
flow. We identify temporal and spatial scales on which the particle trajectory
converges to a spatial Brownian motion. The main technical issue in the proof
is to obtain error estimates for the convergence of the solution of the
stochastic acceleration problem to a momentum diffusion. We also apply our
results to the system of random geometric acoustics equations and show that the
energy density of the acoustic waves undergoes a spatial diffusion.
