Scaling laws for fractional diffusion-wave equations with singular data
type: Item
instance of: scholarly article
title: Scaling laws for fractional diffusion-wave equations with singular data
zbMATH Open document ID: 0970.35174
DOI: 10.1016/S0167-7152(00)00003-1
author: Vo V. Anh
author: Nikolai N. Leonenko
published in: Statistics \& Probability Letters
publication date: 2001-02-14T00:00:00Z
review text: Let \(u(t,x,\beta)\), \(t>0,x\in \mathbb R,0<\beta <1\), be the solution of the Cauchy problem for the equation \[ {\partial^{2\beta }u}/{\partial t^{2\beta }}={\partial^2 u}/{\partial x^2} \] [see \textit{W. R. Schneider} and \textit{W. Wyss}, J. Math. Phys. 30, No.~1, 134-144 (1989; Zbl 0692.45004) and \textit{A. N. Kochubej} [Differ. Equations 26, No. 4, 660-670 (1990; Zbl 0712.35049)] with random Gaussian initial data. It is shown, under certain conditions upon the initial random function \(u(0,x,\beta)\), that the finite-dimensional distributions of the random field \[ X_\varepsilon (t,x,\beta)=\varepsilon^{-\gamma }\left[ u\left( \frac{t}{\varepsilon},\frac{x}{\varepsilon^\beta},\beta \right) - C_0\right] \] (the parameters \(\gamma >0\) and \(C_0\) are determined by \(u(0,x,\beta)\)) converge weakly as \(\varepsilon \to 0\). The limit field is found.
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 62M40
Mathematics Subject Classification ID: 60G60
zbMATH DE Number: 1486957
zbMATH Keywords: fractional diffusion
zbMATH Keywords: Gaussian process
zbMATH Keywords: rescaled solution
reviewed by: Anatoly N. Kochubei
MaRDI profile type: Publication
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full work available at URL: s0167-7152(00)00003-1
OpenAlex ID: W2069521089
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description: scientific article; zbMATH DE number 1486957