Theory of Probability and Mathematical Statistics
Multi-scaling Limits for Time-Fractional Relativistic Diffusion Equations with Random Initial Data
G.-R. Liu, N.-R. Shieh
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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 time-fractional relativistic diffusion equation with the time-fractional parameter $\beta\in(0,1)$, the spatial-fractional parameter $\alpha\in (0,2)$ and the mass parameter $\mathfrak{m}> 0$, subject to random initial data $u(0,\cdot)$ which is characterized as a subordinated Gaussian field. Compared with \cite{AnhHomo} 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 tore-scale the initial data $u_{0}$. In the two scalings, the parameters $\alpha$ and $\mathfrak{m}$ 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.
Keywords: Large-scale limits; Small-scale limits; Relativistic diffusion equations; Random initial data; Multiple It$\hat{\textup{o}}$-Wiener integrals; Subordinated Gaussian fields; Hermite ranks.
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