  The grey incomplete gamma distributions was established by one of the authors
in a previous publication. In this article we use the Kondratiev
characterization theorem to identify those via a suitable Laplace transform
with holomorphic functions with suitable properties. We establish theorems for
the integration and convergence of sequences of these distributions. As direct
applications of these analytic tools we give the examples of Donsker's delta
function, the local time, identify the time-derivative of the process as a
suitable distribution and define the Gamma grey Ornstein-Uhlenbeck process.
