Geometric stable processes and related fractional differential equations

We are interested in the differential equations satisfied by the density of the Geometric Stable processes {G(alpha)(beta)(t); t >= 0}, with stability index alpha is an element of (0; 2] and symmetry parameter beta is an element of [-1; 1], both in the univariate and in the multivariate cases. We resort to their representation as compositions of stable processes with an independent Gamma subordinator. As a preliminary result, we prove that the latter is governed by a differential equation expressed by means of the shift operator. As a consequence, we obtain the space-fractional equation satisfied by the transition density of G(alpha)(beta)(t): For some particular values of alpha and beta; we get some interesting results linked to well-known processes, such as the Variance Gamma process and the first passage time of the Brownian motion.