Supports of Measures in a free additive convolution semigroup

Abstract

In this paper, we study the supports of measures in the free additive convolution semigroup $\{{\mu }^{⊞t}:t>1\}$, where $\mu$ is a Borel probability measure on $\mathbb{R}$. We give a formula for the density of the absolutely continuous part of ${\mu }^{⊞t}$ and use this formula to obtain certain regularizing properties of ${\mu }^{⊞t}$. We show that the number $n(t)$ of the components in the support of ${\mu }^{⊞t}$ is a decreasing function of $t$ and give equivalent conditions so that $n(t)=1$ for sufficiently large $t$. Moreover, a measure $\mu$ so that ${\mu }^{⊞t}$ has infinitely many components in the support for all $t>1$ is given.

21 pages

References in corpus (1)

• Rate of Convergence in the entropic free CLT

Cited by in corpus (1)

• Supports, regularity, and $⊞$-infinite divisibility for measures of the form $({\mu }^{⊞p}{)}^{\uplus q}$