Complex analysis methods in noncommutative probability

Abstract

We study convolutions that arise from noncommutative probability theory. In the case of the free convolutions, we prove that the absolutely continuous part, with respect to the Lebesgue measure, of the free convolution of two probability measures is always nonzero, and has a locally analytic density. Under slightly less general hypotheses, we show that the singular continuous part of the free additive convolution of two probability measures is zero. We also show that any probability measure belongs to a partially defined one-parameter free convolution semigroup. In this context, we find a connection between free and boolean infinite divisibility. For monotonic convolutions, we prove that any infinitely divisible probability measure with respect to monotonic additive or multiplicative convolution belongs to a one-parameter semigroup with respect to the corresponding convolution. Our main tools are several subordination and inversion theorems for analytic functions defined in the upper half-plane. We prove these theorems using the theory of Denjoy-Wolff fixed points.