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, identify the time-derivative of the process as a suitable distribution and define the Gamma grey Ornstein-Uhlenbeck process.