Vol. 305, No. 2, 2020

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Freeness characterizations on free chaos spaces

Solesne Bourguin and Ivan Nourdin

Vol. 305 (2020), No. 2, 447–472
DOI: 10.2140/pjm.2020.305.447
Abstract

We deal with characterizing the freeness and asymptotic freeness of free multiple integrals with respect to a free Brownian motion or a free Poisson process. We obtain three characterizations of freeness, in terms of contraction operators, covariance conditions, and free Malliavin gradients. We show how these characterizations can be used in order to obtain limit theorems, transfer principles, and asymptotic properties of converging sequences.

Keywords
free probability, Wigner integrals, free Poisson integrals, free Malliavin calculus, characterization of freeness, free fourth movement theorems
Mathematical Subject Classification 2010
Primary: 46L54
Secondary: 68H07, 60H30
Milestones
Received: 14 May 2017
Revised: 8 November 2019
Accepted: 9 November 2019
Published: 29 April 2020
Authors
Solesne Bourguin
Department of Mathematics and Statistics
Boston University
Boston, MA
United States
Ivan Nourdin
Department of Mathematics
University of Luxembourg
Esch-sur-Alzette
Luxembourg