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Transactions of the American Mathematical Society

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

 
 

 

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


Authors: Gi-Ren Liu and Narn-Rueih Shieh
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
Published electronically: November 24, 2014
MathSciNet review: 3314812
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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 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

DOI: https://doi.org/10.1090/S0002-9947-2014-06498-2
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): 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
Article copyright: © Copyright 2014 American Mathematical Society

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