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

Let u(t, x), t>0, x ∈ ℝⁿ, be the spatial-temporal random field arising from the solution of a relativistic diffusion equation with the spatialfractional parameter α ∈ (0, 2) and the mass parameter m > 0, subject to a random initial condition u(0, 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, x). Both the Gaussian and the non-Gaussian limit theorems are discussed. The smallscale scaling involves not only scaling on u(t, x) but also re-scaling the initial data; this is a new type result for the literature. Moreover, in the two scalings the parameter α ∈ (0, 2) and the parameter m > 0 play distinct roles for the scaling and the limiting procedures.