Multi-variable subordination distributions for free additive convolution

Let k be a positive integer and let Dcpkq 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 P Dcpkq, where ` is the operation of free additive convolution on Dcpkq. 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 Dcpkq 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.