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 α(t); t ≥ 0 } , with stability index α ∈ (0, 2] and symmetry parameter β ∈ [−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 α(t). For some particular values of α and β, we get some interesting results linked to wellknown processes, such as the Variance Gamma process and the first passage time of the Brownian motion.