Centre for Mathematical Physics and Stochastics

Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

MPS-RR 2003-7

March 2003

In this paper we continue our studies, initiated in \cite{bt1},\cite{bt2} and \cite{bt3}, of the connections between the classes of infinitely divisible probability measures in classical and in free probability. We show that the free cumulant transform of any freely infinitely divisible probability measure equals the classical cumulant transform of a certain classically infinitely divisible probability measure, and we give several characterizations of the latter measure, including an interpretation in terms of stochastic integration. We find, furthermore, an alternative definition of the Bercovici-Pata bijection, which passes directly from the classical to the free cumulant transform, without passing through the Lévy-Khintchine representations (classical and free, respectively). As a byproduct, of some independent interest, the derivation in the final section establishes the existence of a one-to-one mapping of the class of Levy measures into a subset of that class, whose elements have densities, the restrictions to $]-\infty,0[$ and $]0,\infty[$ of which are representable as Laplace transforms.

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