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