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Space–Time Duality for Fractional Diffusion

Part of: Stochastic processes Real functions

Published online by Cambridge University Press:  14 July 2016

Boris Baeumer ,
Mark M. Meerschaert  and
Erkan Nane
Boris Baeumer*
Affiliation:
University of Otago
Mark M. Meerschaert*
Affiliation:
Michigan State University
Erkan Nane*
Affiliation:
Auburn University
*
∗Postal address: Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, New Zealand. Email address: bbaeumer@maths.otago.ac.nz
∗∗Postal address: Department of Probability and Statistics, Michigan State University, East Lansing, MI 48823, USA. Email address: mcubed@stt.msu.edu
∗∗∗Postal address: 221 Parker Hall, Department of Mathematics and Statistics, Auburn University, Auburn, Al 36849, USA. Email address: nane@auburn.edu
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Abstract

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Zolotarev (1961) proved a duality result that relates stable densities with different indices. In this paper we show how Zolotarev's duality leads to some interesting results on fractional diffusion. Fractional diffusion equations employ fractional derivatives in place of the usual integer-order derivatives. They govern scaling limits of random walk models, with power-law jumps leading to fractional derivatives in space, and power-law waiting times between the jumps leading to fractional derivatives in time. The limit process is a stable Lévy motion that models the jumps, subordinated to an inverse stable process that models the waiting times. Using duality, we relate the density of a spectrally negative stable process with index 1<α<2 to the density of the hitting time of a stable subordinator with index 1/α, and thereby unify some recent results in the literature. These results provide a concrete interpretation of Zolotarev's duality in terms of the fractional diffusion model. They also illuminate a current controversy in hydrology, regarding the appropriate use of space- and time-fractional derivatives to model contaminant transport in river flows.

Keywords

Limit theorystable processsubordinatorfractional diffusionduality

MSC classification

Secondary: 60G52: Stable processes 26A33: Fractional derivatives and integrals
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009 , pp. 1100 - 1115
Copyright
Copyright © Applied Probability Trust 2009 

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