## 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.## References

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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