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

Abstract

Let u ( t , x ) , t > 0 , x ∈ R n , 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 α ∈ ( 0 , 2 ) \alpha \in (0,2) and the mass parameter m > 0 \mathfrak {m}> 0 , subject to a random initial condition u ( 0 , x ) 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 , x ) 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 , x ) 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 α ∈ ( 0 , 2 ) \alpha \in (0,2) and the parameter m > 0 \mathfrak {m}> 0 play distinct roles for the scaling and the limiting procedures.