Regularity results for free L\'{e}vy processes

TL;DR

This paper characterizes when free additive convolution semigroups combined with a fixed measure are absolutely continuous with positive, analytic densities, and describes the support structure based on the Lévy measure.

Contribution

It provides necessary and sufficient conditions for regularity and analyticity of densities in free convolution semigroups, linking these properties to the Lévy measure.

Findings

Conditions for Lebesgue absolute continuity and positivity of densities.

Criteria for analyticity at zeros of the density.

Finite connected components in support based on Lévy measures.

Abstract

Given a free additive convolution semigroup $\left(\mu_t\right)_{t\geq 0}$ and a probability measure $\nu$ on $\mathbb{R}$, we find the necessary and sufficient conditions for the process $\mu_t \boxplus \nu$ to be Lebesgue absolutely continuous with a positive and analytic density throughout $\mathbb{R}$ at all time $t>0$. For semigroups without this property, we find the necessary and sufficient conditions for the density of $\mu_t \boxplus \nu$ to be analytic at its zeros. These results are quantified by the L\'{e}vy measure of the semigroup, making it fairly easy to construct many concrete examples. Finally, we show that $\mu_t \boxplus \nu$ has a finite number of connected components in its support if both the L\'{e}vy measure of $\left(\mu_t\right)_{t \geq 0}$ and the initial law $\nu$ do.