Multi-variable subordination distributions for free additive convolution

TL;DR

This paper explores multi-variable subordination distributions in free additive convolution, using combinatorial and operator methods to connect subordination with infinite divisibility and free Brownian motion.

Contribution

It introduces a new framework for understanding multi-variable subordination distributions and their relation to free infinite divisibility and free Brownian motion.

Findings

Established properties of R-transforms for joint distributions.

Connected subordination distributions to a process of free infinite divisibility.

Provided insights into the relationship between subordination and free Brownian motion.

Abstract

Let k be a positive integer and let D_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 \mu \boxplus \nu with respect to \nu" for \mu, \nu \in D_k, where \boxplus is the operation of free additive convolution on D_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 \boxplus-infinite divisibility on D_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.