  In the context of non-Gaussian analysis, Schneider [27] introduced grey noise
measures, built upon Mittag-Leffler functions; analogously, grey Brownian
motion and its generalizations were constructed (see, for example, [25], [6],
[7], [8]). In this paper, we construct and study a new non-Gaussian measure, by
means of the incomplete-gamma function (exploiting its complete monotonicity).
We label this measure Gamma-grey noise and we prove, for it, the existence of
Appell system. The related generalized processes, in the infinite dimensional
setting, are also defined and, through the use of the Riemann-Liouville
fractional operators, the (possibly tempered) Gamma-grey Brownian motion is
consequently introduced. A number of different characterizations of these
processes are also provided, together with the integro-differential equation
satisfied by their transition densities. They allow to model anomalous
diffusions, mimicking the procedures of classical stochastic calculus.
