Free Brownian motion and free convolution semigroups: multiplicative case

Ping Zhong

Vol. 269 (2014), No. 1, 219–256

DOI: 10.2140/pjm.2014.269.219

Abstract

We consider a pair of probability measures $\mu ,\nu$ on the unit circle such that ${\Sigma }_{\lambda }\left({\eta }_{\nu }\left(z\right)\right)=z∕{\eta }_{\mu }\left(z\right)$. We prove that the same type of equation holds for any $t\ge 0$ when we replace $\nu$ by $\nu ⊠{\lambda }_t$ and $\mu$ by ${\mathbb{M}}_t\left(\mu \right)$, where ${\lambda }_t$ is the free multiplicative analogue of the normal distribution on the unit circle of $ℂ$ and ${\mathbb{M}}_t$ is the map defined by Arizmendi and Hasebe. These equations are a multiplicative analogue of equations studied by Belinschi and Nica. In order to achieve this result, we study infinite divisibility of the measures associated with subordination functions in multiplicative free Brownian motion and multiplicative free convolution semigroups. We use the modified $\mathcal{S}$-transform introduced by Raj Rao and Speicher to deal with the case that $\nu$ has mean zero. The same type of the result holds for convolutions on the positive real line. In the end, we give a new proof for some Biane’s results on the densities of the free multiplicative analogue of the normal distributions.

Keywords

multiplicative free convolution, free Brownian motion, free convolution semigroups

Mathematical Subject Classification 2010

Primary: 46L54

Milestones

Received: 13 October 2012

Revised: 30 August 2013

Accepted: 3 September 2013

Published: 15 July 2014