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