Multi-variable subordination distributions for free additive convolution

Let k be a positive integer and let Dc(k) denote the space of joint distributions for k-tuples of selfadjoint elements in C∗-probability space. The paper studies the concept of “subordination distribution of μ⊞ν with respect to ν” for μ,ν∈Dc(k), where ⊞ is the operation of free additive convolution on Dc(k). The main tools used in this study are combinatorial properties of R-transforms for joint distributions and a related operator model, with operators acting on the full Fock space.Multi-variable subordination turns out to have nice relations to a process of evolution towards ⊞-infinite divisibility on Dc(k) that was recently found by Belinschi and Nica (arXiv: 0711.3787). Most notably, one gets better insight into a connection which this process was known to have with free Brownian motion.