The Lévy–Khintchine formula and Nica–Speicher property for deformations of the free convolution
Volume 78 / 2007
Abstract
We study deformations of the free convolution arising via invertible transformations of probability measures on the real line $T:\mu\mapsto T\mu$. We define new associative convolutions of measures by $$ \mu \mathbin{\mathbin{\boxplus }_T} \nu = T^{-1}(T\,\mu \mathbin{\boxplus } T\,\nu). $$ We discuss infinite divisibility with respect to these convolutions, and we establish a Lévy–Khintchine formula. We conclude the paper by proving that for any such deformation of free probability all probability measures $\mu$ have the Nica–Speicher property, that is, one can find their convolution power $\mu^{\mathbin{\mathbin{\boxplus }_T} s}$ for all $s\ge1$. This behaviour is similar to the free case, as in the original paper of Nica and Speicher [NS].
