Fractional diffusion-type equations with exponential and logarithmic differential operators

TL;DR

This paper extends space-fractional diffusion equations by incorporating exponential and logarithmic differential operators, linking these to stochastic processes like Poisson, gamma subordinated, and birth processes, thus broadening the understanding of fractional diffusion models.

Contribution

It introduces novel fractional diffusion equations with exponential and logarithmic operators and connects them to specific stochastic processes, expanding the theoretical framework of fractional diffusion.

Findings

Addition of exponential operator yields a Poisson process with drift.

Logarithmic operator leads to a gamma subordinated stable process.

Non-linear extension involves time-changing with a birth process.

Abstract

We deal with some extensions of the space-fractional diffusion equation, which is satisfied by the density of a stable process (see Mainardi, Luchko, Pagnini (2001)): the first equation considered here is obtained by adding an exponential differential operator expressed in terms of the Riesz-Feller derivative. We prove that this produces a random additional term 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 a non-linear extension of the space-fractional diffusion equation is satisfied by the transition density of the process obtained by time-changing the stable process with an independent linear birth process with drift.