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Fractional Calculus and Applied Analysis

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Volume 19, Issue 2


A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation

Gianni Pagnini
  • Corresponding author
  • BCAM – Basque Center for Applied Mathematics Alameda de Mazarredo 14, E–48009 Bilbao, Basque Country – SPAIN
  • Ikerbasque - Basque Foundation for Science Calle de Mar´ıa D´ıaz de Haro 3, E–48013 Bilbao, Basque Country – SPAIN
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Paolo Paradisi
  • Corresponding author
  • BCAM – Basque Center for Applied Mathematics Alameda de Mazarredo 14, E–48009 Bilbao, Basque Country – SPAIN
  • ISTI–CNR Istituto di Scienza e Tecnologie dell’ Informazione “A. Faedo” Via Moruzzi 1, I–56124 Pisa, ITALY
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Published Online: 2016-05-04 | DOI: https://doi.org/10.1515/fca-2016-0022


The stochastic solution with Gaussian stationary increments is established for the symmetric space-time fractional diffusion equation when 0 < β < α ≤ 2, where 0 < β ≤ 1 and 0 < α ≤ 2 are the fractional derivation orders in time and space, respectively. This solution is provided by imposing the identity between two probability density functions resulting (i) from a new integral representation formula of the fundamental solution of the symmetric space-time fractional diffusion equation and (ii) from the product of two independent random variables. This is an alternative method with respect to previous approaches such as the scaling limit of the continuous time random walk, the parametric subordination and the subordinated Langevin equation. A new integral representation formula for the fundamental solution of the space-time fractional diffusion equation is firstly derived. It is then shown that, in the symmetric case, a stochastic solution can be obtained by a Gaussian process with stationary increments and with a random wideness scale variable distributed according to an arrangement of two extremal Lévy stable densities. This stochastic solution is self-similar with stationary increments and uniquely defined in a statistical sense by the mean and the covariance structure.

Numerical simulations are carried out by choosing as Gaussian process the fractional Brownian motion. Sample paths and probability densities functions are shown to be in agreement with the fundamental solution of the symmetric space-time fractional diffusion equation.

MSC 2010: Primary 26A33; Secondary 60G20, 60G22, 82C31

Key Words and Phrases: anomalous diffusion; space-time fractional diffusion equation; stochastic solution; Gaussian processes; fractional Brownian motion; self-similar stochastic process; stationary increments

Please cite to this paper as published in: Fract. Calc. Appl. Anal., Vol. 19, No 2 (2016), pp. 408–440, DOI: 10.1515/fca-2016-0022


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About the article

Received: 2015-05-05

Revised: 2015-11-30

Published Online: 2016-05-04

Published in Print: 2016-04-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 19, Issue 2, Pages 408–440, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2016-0022.

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Mathematics, 2019, Volume 7, Number 12, Page 1145
Francesco Di Tullio, Paolo Paradisi, Renato Spigler, and Gianni Pagnini
Frontiers in Physics, 2019, Volume 7
É. C. Rocha, M. G. E. da Luz, E. P. Raposo, and G. M. Viswanathan
Physical Review E, 2019, Volume 100, Number 1
Фатима Гидовна Хуштова and Fatima Gidovna Khushtova
Вестник Самарского государственного технического университета. Серия «Физико-математические науки», 2018, Volume 22, Number 4, Page 774
Oleksii Yu Sliusarenko, Silvia Vitali, Vittoria Sposini, Paolo Paradisi, Aleksei Chechkin, Gastone Castellani, and Gianni Pagnini
Journal of Physics A: Mathematical and Theoretical, 2019, Volume 52, Number 9, Page 095601
Trifce Sandev, Weihua Deng, and Pengbo Xu
Journal of Physics A: Mathematical and Theoretical, 2018, Volume 51, Number 40, Page 405002
Silvia Vitali, Vittoria Sposini, Oleksii Sliusarenko, Paolo Paradisi, Gastone Castellani, and Gianni Pagnini
Journal of The Royal Society Interface, 2018, Volume 15, Number 145, Page 20180282
Vittoria Sposini, Aleksei V Chechkin, Flavio Seno, Gianni Pagnini, and Ralf Metzler
New Journal of Physics, 2018, Volume 20, Number 4, Page 043044
Daniel Molina-García, Tuan Minh Pham, Paolo Paradisi, Carlo Manzo, and Gianni Pagnini
Physical Review E, 2016, Volume 94, Number 5

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