  Let D be the space of non-commutative distributions of k-tuples of
selfadjoints in a C*-probability space (for a fixed k). We introduce a
semigroup of transformations B_t of D, such that every distribution in D
evolves under the B_t towards infinite divisibility with respect to free
additive convolution. The very good properties of B_t come from some special
connections that we put into evidence between free additive convolution and the
operation of Boolean convolution.
  On the other hand we put into evidence a relation between the transformations
B_t and free Brownian motion. More precisely, we introduce a transformation Phi
of D which converts the free Brownian motion started at an arbitrary
distribution m in D into the process B_t (Phi(m)), t>0.
