Multi-scaling limits for time-fractional relativistic diffusion equations with random initial data
Authors:
G.-R. Liu and N.-R. Shieh
Journal:
Theor. Probability and Math. Statist. 95 (2017), 109-130
MSC (2010):
Primary 60G60, 60H05, 62M15; Secondary 35K15
DOI:
https://doi.org/10.1090/tpms/1025
Published electronically:
February 28, 2018
MathSciNet review:
3631647
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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 time-fractional relativistic diffusion equation with the time-fractional parameter $\beta \in (0,1)$, the spatial-fractional parameter $\alpha \in (0,2)$ and the mass parameter $\mathfrak {m}> 0$, subject to random initial data $u(0,\boldsymbol \cdot )$ 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 $\alpha$ and $\mathfrak {m}$ 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.
References
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- V. V. Anh and N. N. Leonenko, Non-Gaussian scenarios for the heat equation with singular initial data, Stoch. Processes Appl. 84 (1999), 91–114. MR 1720100
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- V. V. Anh and N. N. Leonenko, Spectral analysis of fractional kinetic equations with random data, J. Statist. Phys. 104 (2001), 1349–1387. MR 1859007
- V. V. Anh and N. N. Leonenko, Renormalization and homogenization of fractional diffusion equations with random data, Probab. Theory Rel. Fields 124 (2002), 381–408. MR 1939652
- V. V. Anh, N. N. Leonenko, and L. M. Sakhno, Higher-order spectral densities of fractional random fields, J. Statist. Phys. 26 (2003), 789–814. MR 1972127
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- G.-R. Liu and N.-R. Shieh, Multi-scaling limits for relativistic diffusion equations with random initial data, Trans. Amer.Math. Soc. 367 (2015), no. 5, 3423–3446. MR 3314812
- P. Major, Muliple Wiener–Itô Integrals, Lect. Note in Math., vol. 849, Springer, 1981. MR 611334
- M. Rosenblatt, Remark on the Burgers equation, J. Math. Phys. 9 (1968), 1129–1136. MR 0264252
- M. D. Ruiz-Medina, J. M. Angulo, and V. V. Anh, Scaling limit solution of the fractional Burgers equation, Stoch. Process. Appl. 93 (2001), 285–300. MR 1828776
- M. Ryznar, Estimates of Green function for relativistic $\alpha$-stable process, Potential Anal. 17 (2002), 1–23. MR 1906405
- N.-R. Shieh, On time-fractional relativistic diffusion equations, J. Pseudo-Differ. Oper. Appl. 3 (2012), 229–237. MR 2925182
- N.-R. Shieh, Free fields associated with the relativistic operator $(\mathfrak {m}-\sqrt {\mathfrak {m}^{2}-\Delta })$, J. Pseudo-Differ. Oper. Appl. 3 (2012), 309–319. MR 2964808
- M. S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrsch. verw. Geb. 50 (1979), 53–83. MR 550123
- M. W. Wong, A contraction semigroup generated by a pseudo-differential operator, Diff. and Int. Eq. 5 (1992), 193–200. MR 1141736
- M. W. Wong, An Introduction to Pseudo-Differential Operators, 2nd Edition, World Scientific, 1999. MR 1698573
- W. Woyczyński, Burgers-KPZ turbulence, Göttingen Lectures, Lect. Note in Math., vol. 1700, Springer, 1998. MR 1732301
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Additional Information
G.-R. Liu
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan
Email:
girenliu@mail.ncku.edu.tw
N.-R. Shieh
Affiliation:
Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan
Email:
shiehnr@ntu.edu.tw
Keywords:
Large-scale limits,
small-scale limits,
relativistic diffusion equations,
random initial data,
multiple Itô–Wiener integrals,
subordinated Gaussian fields,
Hermite ranks
Received by editor(s):
September 23, 2016
Published electronically:
February 28, 2018
Additional Notes:
The first author was supported in part by NCTS/TPE and the Taiwan Ministry of Science and Technology under Grant MOST 104-2115-M-006-016-MY2. This research also received funding from the Headquarter of University Advancement at National Cheng Kung University, which is sponsored by the Ministry of Education, Taiwan, ROC
Dedicated:
This paper is dedicated to the 65th birthday of Professor Nikolai Leonenko
Article copyright:
© Copyright 2018
American Mathematical Society