Multi-scaling Limits for Relativistic Diffusion Equations with Random Initial Data

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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 spatial-fractional 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 small-scale scaling involves not only to scale on u(t,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 α∈ (0,2) and the parameter m> 0 paly distinct roles for the scaling and the limiting procedures.