Positivity of the fundamental solution for fractional diffusion and wave equations

We study the question of positivity of the fundamental solution for fractional diffusion and wave equations of the form, which may be of fractional order both in space and time. We give a complete characterization for the positivity of the fundamental solution in terms of the order of the time derivative $\alpha\in(0,2)$, the order of the spatial derivative $\beta\in (0,2]$ and the spatial dimension $d$. It turns out that the fundamental solution fails to be positive for all $\alpha\in (1,2)$, and either $\beta\in (0,2]$ and $d\ge 2$ or $\beta<\alpha$ and $d=1$, whereas in the other cases it remains positive. The proof is based on delicate properties of the Fox H-functions and the Mittag-Leffler functions.