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

G. R. Liu, N. R. Shieh

Research output: Contribution to journalArticle

Abstract

Let u(t, x), t > 0, x ∈ ℝn, be the spatial-temporal random field arising from the solution of a time-fractional relativistic diffusion equation with the time-fractional parameter β ∈ (0, 1), the spatial-fractional parameter α ∈ (0, 2) and the mass parameter m > 0, subject to random initial data u(0, ·) 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 u0. In the two scalings, the parameters α and 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.

Original languageEnglish
Pages (from-to)109-130
Number of pages22
JournalTheory of Probability and Mathematical Statistics
Volume95
DOIs
Publication statusPublished - 2017 Jan 1

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Multiscaling
Scaling Limit
Diffusion equation
Fractional
Scaling
Limiting
Gaussian Fields
Covariance Structure
Heat Kernel
Random Field
Decay
Distinct

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

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Multi-scaling limits for time-fractional relativistic diffusion equations with random initial data. / Liu, G. R.; Shieh, N. R.

In: Theory of Probability and Mathematical Statistics, Vol. 95, 01.01.2017, p. 109-130.

Research output: Contribution to journalArticle

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