Summary: Let \(\mathcal M\) denote the space of Borel probability measures on \(\mathbb R\). For every \(t\geq 0\), we consider the transformation \(\mathbb B_t : {\mathcal M}\to{\mathcal M}\) defined by
\[ \mathbb B_t(\mu)= (\mu^{\boxplus(1+t)})^{+\!\!\!\!\cup(1/(1+t))}, \quad \mu\in{\mathcal M}, \]
where \(\boxplus\) and \(+\!\!\!\!\!\!\bigcup\) are the operations of free additive convolution and, respectively, of Boolean convolution on \(\mathcal M\), and where the convolution powers with respect to \(\boxplus\) and \(+\!\!\!\!\!\!\bigcup\) are defined in the natural way. We show that \(\mathbb B_s\circ\mathbb B_t=\mathbb B_{s+t}\) \(\forall s, t > 0\) and that, quite surprisingly, every \(\mathbb B_t\) is a homomorphism for the operation of free multiplicative convolution \(\boxtimes\) (that is, \(\mathbb B_t(\mu\boxtimes \nu) = \mathbb B_t(\mu)\boxtimes\mathbb B_t(\nu)\) for all \(\mu,\nu\in\mathcal M\) such that at least one of \(\mu,\nu\) is supported on \([0,\infty)\)).
We prove that, for \(t=1\), the transformation \(\mathbb B_1\) coincides with the canonical bijection \(\mathbb B:{\mathcal M}\to {\mathcal M}_{\text{inf-div}}\) discovered by H.Bercovici and V.Pata [Ann.Math.(2) 149, No.3, 1023–1060 (1999; Zbl 0945.46046)] in their study of the relations between infinite divisibility in free and in Boolean probability. Here, \({\mathcal M}_{\text{inf-div}}\) stands for the set of probability distributions in \(\mathcal M\) which are infinitely divisible with respect to the operation \(\boxplus\). As a consequence, we have that \(\mathbb B_t(\mu)\) is \(\boxplus\)-infinitely divisible for every \(\mu\in\mathcal M\) and every \(t\geq1\).
On the other hand, we put into evidence a relation between the transforrnations \(\mathbb B_t\) and the free Brownian motion; indeed, Theorem 1.6 of the paper gives an interpretation of the transformations \(\mathbb B_t\) as a way of recasting the free Brownian motion, where the resulting process becomes multiplicative with respect to \(\boxtimes\), and always reaches \(\boxplus\)-infinite divisibility by the time \(t =1\).