Regularity of distributions of Wigner integrals

TL;DR

This paper investigates the distributional regularity of Wigner integrals in free probability, proving they cannot have atoms and extending free Malliavin calculus with directional gradients to analyze their properties.

Contribution

It introduces directional gradients in free Malliavin calculus, addressing regularity of Wigner integrals and extending methods to exclude atoms and zero-divisors.

Findings

Non-trivial Wigner chaos elements have atomless distributions

Directional gradients bridge free Malliavin calculus and non-commutative derivations

Zero-divisors are excluded in finite Wigner chaos

Abstract

Wigner integrals and the corresponding Wigner chaos were introduced by P. Biane and R. Speicher in 1998 as a non-commutative counterpart of classical Wiener-It\^o integrals and the corresponding Wiener-It\^o chaos, respectively, in free probability. In the classical case, a famous result of I. Shigekawa states that non-trivial elements in the finite Wiener-It\^o chaos have an absolutely continuous distribution. We provide here a first contribution to such regularity questions for Wigner integrals by showing that the distribution of non-trivial elements in the finite Wigner chaos cannot have atoms. This answers a question of I. Nourdin and G. Peccati. For doing so, we establish the notion of directional gradients in the context of the free Malliavin calculus. These directional gradients bridge between free Malliavin calculus and the theory of non-commutative derivations as initiated by D. Voiculescu and Y. Dabrowski. Methods recently invented by R. Speicher, M. Weber, and the author for treating similar questions in the case of finitely many variables are extended, such that they apply to directional gradients. This approach also excludes zero-divisors for the considered elements in the finite Wigner chaos.