### Abstract

Let u(t, x), t>0, x ∈ ℝ^{n}, be the spatial-temporal random field arising from the solution of a relativistic diffusion equation with the spatialfractional parameter α ∈ (0, 2) and the mass parameter m > 0, subject to a random initial condition u(0, 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). Both the Gaussian and the non-Gaussian limit theorems are discussed. The smallscale scaling involves not only scaling on u(t, 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) and the parameter m > 0 play distinct roles for the scaling and the limiting procedures.

Original language | English |
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Pages (from-to) | 3423-3446 |

Number of pages | 24 |

Journal | Transactions of the American Mathematical Society |

Volume | 367 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2015 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*367*(5), 3423-3446. https://doi.org/10.1090/S0002-9947-2014-06498-2

}

*Transactions of the American Mathematical Society*, vol. 367, no. 5, pp. 3423-3446. https://doi.org/10.1090/S0002-9947-2014-06498-2

**Multi-scaling limits for relativistic diffusion equations with random initial data.** / Liu, Gi Ren; Shieh, Narn Rueih.

Research output: Contribution to journal › Article

TY - JOUR

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

AU - Liu, Gi Ren

AU - Shieh, Narn Rueih

PY - 2015/1/1

Y1 - 2015/1/1

N2 - Let u(t, x), t>0, x ∈ ℝn, be the spatial-temporal random field arising from the solution of a relativistic diffusion equation with the spatialfractional parameter α ∈ (0, 2) and the mass parameter m > 0, subject to a random initial condition u(0, 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). Both the Gaussian and the non-Gaussian limit theorems are discussed. The smallscale scaling involves not only scaling on u(t, 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) and the parameter m > 0 play distinct roles for the scaling and the limiting procedures.

AB - Let u(t, x), t>0, x ∈ ℝn, be the spatial-temporal random field arising from the solution of a relativistic diffusion equation with the spatialfractional parameter α ∈ (0, 2) and the mass parameter m > 0, subject to a random initial condition u(0, 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). Both the Gaussian and the non-Gaussian limit theorems are discussed. The smallscale scaling involves not only scaling on u(t, 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) and the parameter m > 0 play distinct roles for the scaling and the limiting procedures.

UR - http://www.scopus.com/inward/record.url?scp=84923261634&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923261634&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2014-06498-2

DO - 10.1090/S0002-9947-2014-06498-2

M3 - Article

AN - SCOPUS:84923261634

VL - 367

SP - 3423

EP - 3446

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -