Supports of Measures in a free additive convolution semigroup

TL;DR

This paper investigates the structure and regularity of measures in a free additive convolution semigroup, providing formulas for densities, support component behavior, and conditions for support connectedness.

Contribution

It introduces a formula for the density of the absolutely continuous part of measures in the semigroup and characterizes the support's component count as a decreasing function of t.

Findings

The density formula for the absolutely continuous part of 0^{oxplus t}

Support components decrease as t increases

Conditions for the support to be connected for large t

Abstract

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