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 ${\int f(x,v)\psi(v) d \mu( v)}$when $f$ and $v\cdot \nabla_xf$ both belong to $L^2$, and the variable $v$ lies in a discrete subset of $\mathbb R^D$.
First of all, we provide a rate, depending on the number of velocities, to the defect of $H^{\frac{1}{2}}$ regularity. 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 investigate the consistency with the diffusion asymptotics of a Monte–Carlo–like discrete velocity model.

In Analysis & PDE