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 |$\{\mu ^{\boxplus t}:t>1\}$|⁠, where μ 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 nonincreasing function of t and give equivalent conditions so that n(t)=1 for sufficiently large t. Moreover, a measure μ so that |$\mu ^{\boxplus t}$| has infinitely many components in the support for all t>1 is given.

Locations

• International Mathematics Research Notices - View
• arXiv (Cornell University) - View - PDF