A Connection between Free and Classical In nite Divisibility

In this paper we continue our studies, initiated in [BT1],[BT2] and [BT3], of the con-nections between the classes of innitely divisible probability measures in classical and in free probability. We show that the free cumulant transform of any freely innitely divisible probability measure equals the classical cumulant transform of a certain classically innitely divisible probability measure, and we give several characterizations of the latter measure, including an interpretation in terms of stochastic integration. We nd, furthermore, an al-ternative denition of the Bercovici-Pata bijection, which passes directly from the classical to the free cumulant transform, without passing through the Levy-Khintchine representa-tions (classical and free, respectively). As a byproduct, of some independent interest, the derivation in the nal 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 den-sities, the restrictions to]−1; 0 [ and]0;1 [ of which are representable as Laplace transforms. 1 Introduction