Fractional diffusion-type equations with exponential and logarithmic differential operators

Luisa Beghin

Stochastic Processes and their Applications, 2018, vol. 128, issue 7, 2427-2447

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.

Keywords: Fractional exponential operator; Fractional logarithmic operator; Riesz–Feller derivative; Gamma process with drift; Yule–Furry process (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:128:y:2018:i:7:p:2427-2447

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DOI: 10.1016/j.spa.2017.09.013

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