REGULARITY OF VELOCITY AVERAGES FOR TRANSPORT EQUATIONS ON RANDOM DISCRETE VELOCITY GRIDS

We go back to the question of the regularity of the "velocity average" integral f (x, upsilon)psi(upsilon)d mu(upsilon) when f and upsilon.del(x) f both belong to L-2, and the variable upsilon lies in a discrete subset of R-D. First of all, we provide a rate, depending on the number of velocities, for the defect of H-1/2 regularity which is reached when v ranges over a continuous set. Second of all, we show that the H-1/2 regularity holds in expectation when the set of velocities is chosen randomly. We apply this statement to investigate the consistency with the diffusion asymptotics of a Monte Carlo-like discrete velocity model.