Free Brownian motion and free convolution semigroups: multiplicative case

Abstract

We consider a pair of probability measures $\mu ,\nu$ on the unit circle such that ${\Sigma }_{\lambda }({\eta }_{\nu }(z))=z/{\eta }_{\mu }(z)$. 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 ${\mathbb{M}}_t(\mu )$, where ${\lambda }_t$ is the free multiplicative analogue of the normal distribution on the unit circle of $\mathbb{C}$ 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. We also obtain some regularity properties for the free multiplicative analogue of the normal distributions.

to appear in Pacific Journal of Mathematics

References in corpus (2)

• Semigroups related to additive and multiplicative, free and Boolean convolutions
• Asymptotic Expansions in the CLT in Free Probability

Cited by in corpus (3)

• On the free convolution with a free multiplicative analogue of the normal distribution
• On the support of measures in multiplicative free convolution semigroups
• Local limit theorems for multiplicative free convolutions