Fractional diffusion-type equations with exponential and logarithmic differential operators

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.