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Continuous-time random walk and parametric subordination in fractional diffusion

Author

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  • Gorenflo, Rudolf
  • Mainardi, Francesco
  • Vivoli, Alessandro

Abstract

The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW) is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit, we obtain a (generally non-Markovian) diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination, applied to a combination of a Markov process with a positively oriented Lévy process, we generate and display sample paths for some special cases.

Suggested Citation

  • Gorenflo, Rudolf & Mainardi, Francesco & Vivoli, Alessandro, 2007. "Continuous-time random walk and parametric subordination in fractional diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 34(1), pages 87-103.
    • Handle: RePEc:eee:chsofr:v:34:y:2007:i:1:p:87-103
    DOI: 10.1016/j.chaos.2007.01.052
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    References listed on IDEAS

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    Cited by:

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    2. Agrawal, S.K. & Srivastava, M. & Das, S., 2012. "Synchronization of fractional order chaotic systems using active control method," Chaos, Solitons & Fractals, Elsevier, vol. 45(6), pages 737-752.
    3. Golder, J. & Joelson, M. & Néel, M.C., 2011. "Mass transport with sorption in porous media," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(10), pages 2181-2189.
    4. Hang Yu & Chenhui Zhu & Lu Yao & Yan Ma & Yang Ni & Shenkai Li & Huan Li & Yang Liu & Yuming Wang, 2023. "The Two Stage Moisture Diffusion Model for Non-Fickian Behaviors of 3D Woven Composite Exposed Based on Time Fractional Diffusion Equation," Mathematics, MDPI, vol. 11(5), pages 1-12, February.
    5. Villarroel, Javier & Montero, Miquel, 2009. "On properties of continuous-time random walks with non-Poissonian jump-times," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 128-137.
    6. Carlos Fuentes & Fernando Alcántara-López & Antonio Quevedo & Carlos Chávez, 2021. "Fractional Vertical Infiltration," Mathematics, MDPI, vol. 9(4), pages 1-14, February.
    7. Tawfik, Ashraf M. & Elkamash, I.S., 2022. "On the correlation between Kappa and Lévy stable distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 601(C).
    8. Javier Villarroel & Miquel Montero, 2008. "On properties of Continuous-Time Random Walks with Non-Poissonian jump-times," Papers 0812.2148, arXiv.org.
    9. Dexter Cahoy, 2012. "Moment estimators for the two-parameter M-Wright distribution," Computational Statistics, Springer, vol. 27(3), pages 487-497, September.
    10. Mura, A. & Taqqu, M.S. & Mainardi, F., 2008. "Non-Markovian diffusion equations and processes: Analysis and simulations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(21), pages 5033-5064.

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