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Space–Time Duality for Fractional Diffusion

  • Boris Baeumer (a1), Mark M. Meerschaert (a2) and Erkan Nane (a3)
Abstract

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.

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Copyright
Corresponding author
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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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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