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Fractional diffusion and fractional heat equation

Part of:Markov processesStochastic processes

Published online by Cambridge University Press:  01 July 2016

J. M. Angulo,
M. D. Ruiz-Medina,
V. V. Anh and
W. Grecksch
J. M. Angulo*
Affiliation:
University of Granada
M. D. Ruiz-Medina*
Affiliation:
University of Granada
V. V. Anh*
Affiliation:
Queensland University of Technology
W. Grecksch*
Affiliation:
University of Halle-Wittenberg
*
∗ Postal address: Department of Statistics and Operations Research, University of Granada, Campus Fuente Nueva s/n, E-18071 Granada, Spain.
∗ Postal address: Department of Statistics and Operations Research, University of Granada, Campus Fuente Nueva s/n, E-18071 Granada, Spain.
∗∗ Postal address: Centre in Statistical Science and Industrial Mathematics, Queensland University of Technology, GPO Box 2434, Brisbane, Q. 4001, Australia. Email address: v.anh@fsc.qut.edu.au
∗∗ Postal address: Centre in Statistical Science and Industrial Mathematics, Queensland University of Technology, GPO Box 2434, Brisbane, Q. 4001, Australia. Email address: v.anh@fsc.qut.edu.au
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Abstract

This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.

Keywords

Diffusion processesstochastic heat equationBessel potentialRiesz potential

MSC classification

Primary:60J60: Diffusion processes60G60: Random fields

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 4 , December 2000, pp. 1077 - 1099
Copyright
Copyright © Applied Probability Trust 2000 

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