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 Lévy basis) is studied in detail in terms of the free Lévy-Khintchine representation and a theory parallel to the classical work of Rajput and Rosinski is developed. Finally a Lévy-Itô type decomposition for general free Lévy bases is established.
Interlacing sequences resulting from an interval split-merge dynamics and the induced probability measures
Institut de Mathématiques de Toulouse, Université de Toulouse（图卢兹数学研究所，图卢兹大学；法国国家科学研究中心）；CNRS（康奈尔大学运筹信息与工程学院）；School of Operations Research and Information Engineering, Cornell University
Integral transforms related to Nevanlinna-Pick functions from an analytic, probabilistic and free-probability point of view
我们建立了Nevanlinna-Pick函数类和与谱负Lévy过程相关的指数之间的一个新的联系。因此，我们计算了一些双曲函数的相关特征，并显示了时间完全单调性的性质，类似于Bertoin \& Yor（{ \it On subordinators，self-similar Markov processes and some factorizations of the exponential variable}，Elect.在Probab.，第6卷，第100页。95- 106，2001）的自相似马尔可夫过程。更确切地说，我们显示了一个显着的事实，即从属$ξ$，功能$t \mapsto t^n \, \er[ξ_t^{-p}]$是，取决于指数值$n=0,1,2,\; p>-1$，或伯恩斯坦函数或完全单调函数。特别地，$ξ$是一个谱负Lévy过程的逆时间从属子，当且仅当对于某个$\,p\geq 1$，函数$t \mapsto t \, \er[ξ_t^{-p}]$是一个斯蒂尔切斯变换。最后，我们澄清在何种程度上Nevanlinna-Pick函数相关的自由概率和Voiculescu变换，我们提供了一个反演过程。
