On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution

Let M denote the space of Borel probability measures on R. For every t >= 0 we consider the transformation B-t : M -> M defined by B-t(mu) = (mu(boxed plus(1+t)))((sic)(1/(1+t))), mu is an element of M, where boxed plus and (sic) are the operations of free additive convolution and respectively of Boolean convolution on M, and where the convolution powers with respect to boxed plus and (sic) are defined in the natural way. We show that Bs circle B-t = Bs+t, for all s, t >= 0 and that, quite surprisingly, every B-t is a homomorphism for the operation of free multiplicative convolution boxed plus (that is, B-t (mu boxed plus nu) = B-t (mu) boxed plus B-t (nu) for all mu, nu is an element of M such that at least one of p, v is supported on [0, infinity)). We prove that for t = 1 the transformation B-1 coincides with the canonical bijection B : M -> Minf-div discovered by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. Here Minf-div stands for the set of probability distributions in M which are infinitely divisible with respect to the operation boxed plus. As a consequence, we have that B-t(mu) is boxed plus-infinitely divisible for every p is an element of N and every t >= 1. On the other hand we put into evidence a relation between the transformations B-t and the free Brownian motion; indeed, Theorem 1.6 of the paper gives an interpretation of the transformations B-t as a way of recasting the free Brownian motion, where the resulting process becomes multiplicative with respect to boxed plus, and always reaches boxed plus-infinite divisibility by the time t = 1.