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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Multi-scaling limits for relativistic diffusion equations with random initial data
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by Gi-Ren Liu and Narn-Rueih Shieh PDF
Trans. Amer. Math. Soc. 367 (2015), 3423-3446 Request permission

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 relativistic diffusion equation with the spatial-fractional parameter $\alpha \in (0,2)$ and the mass parameter $\mathfrak {m}> 0$, subject to a random initial condition $u(0,\mathbf {x})$ 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 $u(t,\mathbf {x})$. Both the Gaussian and the non-Gaussian limit theorems are discussed. The small-scale scaling involves not only scaling on $u(t,\mathbf {x})$ but also re-scaling the initial data; this is a new type result for the literature. Moreover, in the two scalings the parameter $\alpha \in (0,2)$ and the parameter $\mathfrak {m}> 0$ play distinct roles for the scaling and the limiting procedures.
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Additional Information
  • Gi-Ren Liu
  • Affiliation: Ph.D. Class in Mathematics, National Taiwan University, Taipei 10617, Taiwan
  • Narn-Rueih Shieh
  • Affiliation: Department of Mathematics, Honorary Faculty, National Taiwan University, Taipei 10617, Taiwan
  • Email: shiehnr@ntu.edu.tw
  • Received by editor(s): July 7, 2012
  • Received by editor(s) in revised form: October 1, 2012, and May 7, 2013
  • Published electronically: November 24, 2014
  • Additional Notes: The first author was partially supported by a Taiwan NSC grant for graduate students
  • © Copyright 2014 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 367 (2015), 3423-3446
  • MSC (2010): Primary 60G60, 60H05, 62M15, 35K15
  • DOI: https://doi.org/10.1090/S0002-9947-2014-06498-2
  • MathSciNet review: 3314812