Self-decomposability and Lévy processes in free probability

Abstract

In this paper we study the bijection, introduced by Bercovici and Pata, between the classes of infinitely divisible probability measures in classical and in free probability. We prove certain algebraic and topological properties of that bijection (in the present paper denoted Λ), and those properties are then used to show, in particular, that Λ maps the class of classically self-decomposable probability measures onto the natural free counterpart which we define here. Further, we study Lévy processes in free probability and use the properties of Λ to construct stochastic integrals with respect to such processes. In particular, we derive the free analogue of the integral representation of self-decomposable random variables.