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 ${\int f(x,v)\psi(v) d \mu( v)}$ when $v\cdot \nabla_xf$ both belong to $L^p$, $p \in [1, \infty)$ and the measured set of velocities $(\mathscr{V}, d \mu)$ satisfy a nondegenerancy assumption. We are interested in the case when the variable $v$ lies in a discrete subset of $\mathbb{R}^D$. We present results obtained in collaboration with T. Goudon. First of all, we provide a rate, depending on the number of velocities, to the defect of $H^{\frac{1}{2}}$ regularity which is reached when $v$ ranges over a continuous set. Second of all, we show that the $H^{\frac{1}{2}}$ 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.

In CEDRAM, Séminaire Laurent Schwartz — EDP et applications