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

Abstract:

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 scaling on $u(t,\mathbf {x})$ but also re-scaling 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 $\mathfrak {m}> 0$ play distinct roles for the scaling and the limiting procedures.