Random integral representations for free-infiniteley divisible and tempered distributions

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.}