### Abstract

Let u(t, x), t > 0, x ∈ ℝ^{n}, 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 u_{0}. 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.

Original language | English |
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Pages (from-to) | 109-130 |

Number of pages | 22 |

Journal | Theory of Probability and Mathematical Statistics |

Volume | 95 |

DOIs | |

Publication status | Published - 2017 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

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*Theory of Probability and Mathematical Statistics*, vol. 95, pp. 109-130. https://doi.org/10.1090/tpms/1025

**Multi-scaling limits for time-fractional relativistic diffusion equations with random initial data.** / Liu, G. R.; Shieh, N. R.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Multi-scaling limits for time-fractional relativistic diffusion equations with random initial data

AU - Liu, G. R.

AU - Shieh, N. R.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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.

AB - 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=85043331306&partnerID=8YFLogxK

U2 - 10.1090/tpms/1025

DO - 10.1090/tpms/1025

M3 - Article

AN - SCOPUS:85043331306

VL - 95

SP - 109

EP - 130

JO - Theory of Probability and Mathematical Statistics

JF - Theory of Probability and Mathematical Statistics

SN - 0094-9000

ER -