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Time-fractional telegraph equations and telegraph processes with brownian time
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  • Published: 12 November 2003

Time-fractional telegraph equations and telegraph processes with brownian time

  • Enzo Orsingher1 &
  • Luisa Beghin1 

Probability Theory and Related Fields volume 128, pages 141–160 (2004)Cite this article

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Abstract

We study the fundamental solutions to time-fractional telegraph equations of order 2α. We are able to obtain the Fourier transform of the solutions for any α and to give a representation of their inverse, in terms of stable densities. For the special case α=1/2, we can show that the fundamental solution is the distribution of a telegraph process with Brownian time. In a special case, this becomes the density of the iterated Brownian motion, which is therefore the fundamental solution to a fractional diffusion equation of order 1/2 with respect to time.

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Authors and Affiliations

  1. Università di Roma ‘‘La Sapienza’‘, p.le Aldo Moro 5, 00185, Roma, Italy

    Enzo Orsingher & Luisa Beghin

Authors
  1. Enzo Orsingher
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  2. Luisa Beghin
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Corresponding author

Correspondence to Enzo Orsingher.

Additional information

This research has been partially supported by the NATO grant No. SA (PST.CLG.976361) 5437.

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Orsingher, E., Beghin, L. Time-fractional telegraph equations and telegraph processes with brownian time. Probab. Theory Relat. Fields 128, 141–160 (2004). https://doi.org/10.1007/s00440-003-0309-8

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  • Received: 05 January 2002

  • Revised: 27 September 2003

  • Published: 12 November 2003

  • Issue Date: January 2004

  • DOI: https://doi.org/10.1007/s00440-003-0309-8

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Keywords

  • Telegraph Equation
  • Fractional-Derivatives
  • Stable Laws
  • Fractional Heat
  • Wave Equations
  • Iterated Brownian Motion
  • Mittag-Leffler Function
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