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” [math] when [math] and [math] both belong to [math] , and the variable [math] lies in a discrete subset of [math] . First of all, we provide a rate, depending on the number of velocities, for the defect of [math] regularity which is reached when [math] ranges over a continuous set. Second of all, we show that the [math] 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.