从解析、概率和自由概率观点看Nevanlinna-Pick函数的积分变换
Integral transforms related to Nevanlinna-Pick functions from an analytic, probabilistic and free-probability point of view
摘要：
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。
英文摘要：
We establish a new connection between the class of Nevanlinna-Pick functions and the one of the exponents associated to spectrally negative Lévy processes. As a consequence, we compute the characteristics related to some hyperbolic functions and we show a property of temporal complete monotonicity, similar to the one obtained via the Lamperti transformation by Bertoin \& Yor ({\it On subordinators, self-similar Markov processes and some factorizations of the exponential variable}, Elect. Comm. in Probab., vol. 6, pp. 95--106, 2001) for self-similar Markov processes. More precisely, we show the remarkable fact that for a subordinator $ξ$, the function $t \mapsto t^n \, \er[ξ_t^{-p}]$ is , depending on the values of the exponents $n=0,1,2,\; p>-1$, or a Bernstein function or a completely monotone function. In particular, $ξ$ is the inverse time subordinator of a spectrally negative Lévy process, if, and only if, for some $\,p\geq 1$, the function $t \mapsto t \, \er[ξ_t^{-p}]$ is a Stieltjes transform. Finally, we clarify to which extent Nevanlinna-Pick functions are related to free-probability and to Voiculescu transforms, and we provide an inversion procedure.
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