Multi-scaling limits for time-fractional relativistic diffusion equations with random initial data

Let $u(t,\mathbf {x})$, $t>0$, $\mathbf {x}\in \mathbb {R}^{n}$, be the spatial-temporal random field arising from the solution of a time-fractional relativistic diffusion equation with the time-fractional parameter $\beta \in (0,1)$, the spatial-fractional parameter $\alpha \in (0,2)$ and the mass parameter $\mathfrak {m}> 0$, subject to random initial data $u(0,\boldsymbol \cdot )$ which is characterized as a subordinated Gaussian field. Compared with work written by Anh and Leoeneko in 2002, we not only study the large-scale limits of the solution field $u$, but also propose a small-scale scaling scheme, which also leads to the Gaussian and the non-Gaussian limits depending on the covariance structure of the initial data. The new scaling scheme involves not only to scale $u$ but also to re-scale the initial data $u_{0}$. In the two scalings, the parameters $\alpha$ and $\mathfrak {m}$ play distinct roles in the process of limiting, and the spatial dimensions of the limiting fields are restricted due to the slow decay of the time-fractional heat kernel.