Additive Geometric Stable Processes and Related Pseudo-Differential Operators

Additive processes are obtained from Levy ones by relaxing the condition of stationary increments, hence they are spatially (but not temporally) homogeneous. By analogy with the case of time-homogeneous Markov processes, one can de ne an in nitesimal generator, which is, of course, a timedependent operator. Additive versions of stable and Gamma processes have been considered in the literature. We introduce here time-inhomogeneous generalizations of the well-known geometric stable process, de ned by means of time-dependent versions of fractional pseudo-dierential operators of logarithmic type. The local Lévy measures are expressed in terms of Mittag-Leffler functions or H-functions with time-dependent parameters. This article also presents some results about propagators related to additive processes.