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Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

MPS-RR 2001-1
January 2001

Selfdecomposability and Lévy Processes in Free Probability


Ole E. Barndorff-Nielsen

S. Thorbjørnsen


In this paper we study the bijection, introduced by Bercovici and Pata in [BP2], 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 Lambda), and those properties are then used to show, in particular, that Lambda maps the class of classically selfdecomposable probability measures onto the natural free counterpart, that we define here. Further, we study Lévy processes in free probability and use the properties of Lambda to construct stochastic integrals w.r.t. such processes. In particular, we derive the free analogue of the integral representation of selfdecomposable random variables.

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This paper has now been published in Bernoulli 8, 323--366 (2002)