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Fractional diffusion-type equations with exponential and logarithmic differential operators

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  • Beghin, Luisa

Abstract

We deal with some extensions of the space-fractional diffusion equation, which is satisfied by the density of a stable process (see Mainardi et al. (2001)): the first equation considered here is obtained by adding an exponential differential (or shift) operator expressed in terms of the Riesz–Feller derivative. We prove that this produces a random component in the time-argument of the corresponding stable process, which is represented by the so-called Poisson process with drift. Analogously, if we add, to the space-fractional diffusion equation, a logarithmic differential operator involving the Riesz-derivative, we obtain, as a solution, the transition semigroup of a stable process subordinated by an independent gamma subordinator with drift. Finally, we show that an extension of the space-fractional diffusion equation, containing both the fractional shift operator and the Feller integral, is satisfied by the transition density of the process obtained by time-changing the stable process with an independent linear birth process with drift.

Suggested Citation

  • Beghin, Luisa, 2018. "Fractional diffusion-type equations with exponential and logarithmic differential operators," Stochastic Processes and their Applications, Elsevier, vol. 128(7), pages 2427-2447.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:7:p:2427-2447
    DOI: 10.1016/j.spa.2017.09.013
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    1. Kozubowski, Tomasz J. & Panorska, Anna K., 1996. "On moments and tail behavior of v-stable random variables," Statistics & Probability Letters, Elsevier, vol. 29(4), pages 307-315, September.
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    3. Beghin, Luisa & Orsingher, Enzo, 2009. "Iterated elastic Brownian motions and fractional diffusion equations," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1975-2003, June.
    4. Mijena, Jebessa B. & Nane, Erkan, 2015. "Space–time fractional stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3301-3326.
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    6. Mohsen Alipour & Luisa Beghin & Davood Rostamy, 2015. "Generalized Fractional Nonlinear Birth Processes," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 525-540, September.
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    Cited by:

    1. Gajda, Janusz & Beghin, Luisa, 2021. "Prabhakar Lévy processes," Statistics & Probability Letters, Elsevier, vol. 178(C).
    2. Zhang, Meihui & Jia, Jinhong & Zheng, Xiangcheng, 2023. "Numerical approximation and fast implementation to a generalized distributed-order time-fractional option pricing model," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).

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