This paper investigates homomorphisms à la Bercovici–Pata between additive and multiplicative convolutions. We also consider their matricial versions which are associated with measures on the space of Hermitian matrices and on the unitary group. The previous results combined with a matricial model of Benaych–Georges and Cabanal–Duvillard allow us to define and study the large $N$ limit of a new matricial model on the unitary group for free multiplicative Lévy processes.
"Matricial model for the free multiplicative convolution." Ann. Probab. 44 (4) 2427 - 2478, July 2016. https://doi.org/10.1214/15-AOP1024