Free completely random measures

Free probability is a noncommutative probability theory introduced by Voiculescu where the concept of independence of classical probability is replaced by the concept of freeness. An important connection between free and classical infinite divisibility was established by Bercovici and Pata (1999) in form of a bijection, mapping the class of classical infinitely divisible laws into the class of free infinitely divisible laws. A particular class of infinitely divisible laws are the completely random measures introduced by Kingman (1967). In this paper, a free analogous of completely random measures is introduced and, a free Poisson process characterization is provided as well as a representation through a free cumulant transform. Furthermore, some examples are displayed.