Some probabilistic properties of fractional point processes.

In this article, the first hitting times of generalized Poisson processesNf(t), related to Bernštein functionsfare studied. For the space-fractional Poisson processes,Nα(t),t> 0 (corresponding tof=xα), the hitting probabilitiesP{Tαk< ∞} are explicitly obtained and analyzed. The processesNf(t) are time-changed Poisson processesN(Hf(t)) with subordinatorsHf(t) and here we studyand obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the formwhereare generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space–time Poisson process is no longer a renewal process. [ABSTRACT FROM AUTHOR]