Completely random measures and Lévy bases in free probability

This paper develops a theory for completely random measures in the framework of free probability. A general existence result for free completely random measures is established, and in analogy to the classical work of Kingman it is proved that such random measures can be decomposed into the sum of a purely atomic part and a (freely) infinitely divisible part. The latter part (termed a free Levy basis) is studied in detail in terms of the free Levy-Khintchine representation and a theory parallel to the classical work of Rajput and Rosinski is developed. Finally a Levy-Ito type decomposition for general free Levy bases is established.