Completely Random Measures and L\'evy Bases in Free probability
Francesca Collet, Fabrizio Leisen, Steen Thorbj{\o}rnsen
TL;DR
This paper extends the theory of completely random measures to free probability, establishing existence, decomposition, and representation results for free Le9vy bases, paralleling classical probability theory.
Contribution
It introduces a comprehensive framework for free completely random measures, including existence, decomposition, and a Le9vy-Khintchine representation in free probability.
Findings
Existence of free completely random measures established.
Decomposition into atomic and infinitely divisible parts proved.
Le9vy-Itf4 decomposition for free Le9vy bases developed.
Abstract
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 L\'evy basis) is studied in detail in terms of the free L\'evy-Khintchine representation and a theory parallel to the classical work of Rajput and Rosinski is developed. Finally a L\'evy-It\^o type decomposition for general free L\'evy bases is established.
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