Stochastic discrete velocity averaging lemmas and Rosseland approximation

In this note, we investigate some questions around velocity averaging lemmas, a class of results which ensure the regularity of the “velocity average” ∫ f(x, v)ψ(v) dμ(v) when f and v · ∇xf both belong to L, p ∈ [1,∞) and the measured set of velocities (V , dμ) satisfy a nondegeneracy assumption. We are interested in the case when the variable v lies in a discrete subset of R. We present results obtained in collaboration with T. Goudon in [2]. First of all, we provide a rate, depending on the number of velocities, to the defect of H regularity which is reached when v ranges over a continuous set. Second of all, we show that the H regularity holds in expectation when the set of velocities is chosen randomly. We apply this statement to obtain a consistency result for the diffusion limit in the case of the Rosseland approximation.