Additive Geometric Stable Processes and Related Pseudo-Differential Operators

摘要

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 define an infinitesimal generator, which is, of course, a time-dependent 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, defined by means of time-dependent versions of fractional pseudo-differential operators of logarithmic type. The local Levy 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.