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Theory of Probability and Mathematical Statistics

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

Authors: G.-R. Liu and N.-R. Shieh
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 95 (2016).
Journal: Theor. Probability and Math. Statist. 95 (2017), 109-130
MSC (2010): Primary 60G60, 60H05, 62M15; Secondary 35K15
Published electronically: February 28, 2018
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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,\boldsymbol \cdot )$ 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 $ 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.

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Additional Information

G.-R. Liu
Affiliation: Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan

N.-R. Shieh
Affiliation: Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan

Keywords: Large-scale limits, small-scale limits, relativistic diffusion equations, random initial data, multiple It\^o--Wiener integrals, subordinated Gaussian fields, Hermite ranks
Received by editor(s): September 23, 2016
Published electronically: February 28, 2018
Additional Notes: The first author was supported in part by NCTS/TPE and the Taiwan Ministry of Science and Technology under Grant MOST 104-2115-M-006-016-MY2. This research also received funding from the Headquarter of University Advancement at National Cheng Kung University, which is sponsored by the Ministry of Education, Taiwan, ROC
Dedicated: This paper is dedicated to the 65th birthday of Professor Nikolai Leonenko
Article copyright: © Copyright 2018 American Mathematical Society

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