  This paper investigates homomorphisms \`a 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 allows us to define and study the large N
limit of a new matricial model on the unitary group for free multiplicative
L\'evy processes.
