Multi-variable subordination distributions for free additive convolution

Abstract

Let k be a positive integer and let D c ( 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 ∈ D c ( k ) , where ⊞ is the operation of free additive convolution on D c ( 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 D c ( 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.