  Additive processes are obtained from L\'{e}vy 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 L\'{e}vy measures are expressed in terms of Mittag-Leffler functions or
$H$-functions with time-dependent parameters.
