Multi-scaling Limits for 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 relativistic diffusion equation with the spatial-fractional parameter $\alpha\in (0,2)$ and the mass parameter $\mathfrak{m}> 0$, subject to a random initial condition $u(0,\mathbf{x})$ which is characterized as a subordinated Gaussian field. In this article, we study the large-scale and the small-scale limits for the suitable space-time re-scalings of the solution field $u(t,\mathbf{x})$. Both the Gaussian and the non-Gaussian limit theorems are discussed. The small-scale scaling involves not only to scale on $u(t,\mathbf{x})$ but also to re-scale the initial data; this is a new-type result for the literature. Moreover, in the two scalings the parameter $\alpha\in (0,2)$ and the parameter $\mathfrak{m}> 0$ paly distinct roles for the scaling and the limiting procedures.