Multi-scaling limits for time-fractional 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 time-fractional relativistic diffusion equation with the time-fractional parameter β ∈ (0, 1), the spatial-fractional parameter α ∈ (0, 2) and the mass parameter m > 0, subject to random initial data u(0, ·) which is characterized as a subordinated Gaussian field. Compared with work written by Anh and Leoeneko in 2002, we not only study the large-scale limits of the solution field u, but also propose a small-scale scaling scheme, which also leads to the Gaussian and the non-Gaussian limits depending on the covariance structure of the initial data. The new scaling scheme involves not only to scale u but also to re-scale the initial data u₀. In the two scalings, the parameters α and m play distinct roles in the process of limiting, and the spatial dimensions of the limiting fields are restricted due to the slow decay of the time-fractional heat kernel.