On hyperbolic characteristic functions from an analytic and a free-probability point of view (original) (raw)

Ona relation between classical and free infinitely divisible transforms

arXiv: Probability, 2017

We study two ways (levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms. For free-selfdecomposable measures we found the formula (a differential equation) for their background driving transforms. We illustrate our methods on the hyperbolic characteristic functions. As a by-product our approach may produce new formulas for some definite integrals.

Integral transforms related to Nevanlinna-Pick functions from an analytic, probabilistic and free-probability point of view

arXiv (Cornell University), 2021

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 (On subordinators, selfsimilar 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 ↦ t n E[ξ −p t ] 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 ≥ 1, the function t ↦ t E[ξ −p t ] 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.

Cauchy transforms of measures viewed as some functionals of Fourier transforms

In memory of Kazimierz Urbanik ABSTRACT. The Cauchy transform of a positive measure plays an important role in complex analysis and more recently in so-called free probability. We show here that the Cauchy transform restricted to the imaginary axis can be viewed as the Fourier transform of some corresponding measures. Thus this allows the full use of that classical tool. Furthermore, we relate restricted Cauchy transforms to classical compound Poisson measures, exponential mixtures, geometric infinite divisibility and free-infinite divisibility. Finally we illustrate our approach with some examples.

Clifford-Fourier transform on hyperbolic space

Mathematical Methods in the Applied Sciences, 2016

In this paper, we introduce a new generalization of the Helgason-Fourier transform using the angular Dirac operator on both the hyperboloid and unit ball models. The explicit integral kernels of even dimension are derived. Furthermore, we establish the formal generating function of the even dimensional kernels. In the computations, fractional integration plays a key unifying role.

Remarks on the tangent function from an analytic and probability point of view

arXiv (Cornell University), 2020

In this note we prove that the function tan(1/it) is a Voiculescu transform of a free-infinitely divisible distribution, that is, it admits the integral representation associated with Pick functions. Moreover, we found its "counterpart" in the classical infinitely divisible measures expressed by a series of Rademacher variables.

Generalized Hyperbolic Distributions

2002

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