Random integral representations for free-infiniteley divisible and tempered distributions

TL;DR

This paper introduces new integral representations for free-infinite divisible and tempered distributions, revealing connections between free probability and classical Lévy processes, with applications to statistical physics.

Contribution

It develops novel random integral mappings that relate free and classical Lévy measures, and introduces new classes of convolution semigroups with significant theoretical implications.

Findings

Established conditions for mixtures of Lévy spectral measures to remain Lévy measures

Discovered a surprising link between free and classical Lévy-Khintchine formulas

Identified classes of tempered stable measures relevant to physics

Abstract

There are given sufficient conditions under which mixtures of dilations of L\'evy spectral measures, on a Hilbert space, are L\'evy measures again. We introduce some random integrals with respect to infinite dimensional L\'evy processes, which in turn give some integral mappings. New classes (convolution semigroups) are introduced. One of them gives an unexpected relation between the free (Voiculescu) and the classical L\'evy-Khintchine formulae while the second one coincides with tempered stable measures (Mantegna nad Stanley) arisen in statistical physics.}