Random matrix models of stochastic integral type for free infinitely divisible distributions

Abstract

The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the L\'{e}vy measures of the random matrix models considered in Benaych-Georges who introduced the models through their measures. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations of random matrices given by stochastic integrals with respect to matrix-valued L\'{e}vy processes. Examples of these random matrix models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type $\int_{0}^{\infty}e^{-t}d\Psi_{t}^{d}$, $d\geq1$, where $\Psi_{t}^{d}$ is a $d\times d$ matrix-valued L\'{e}vy process satisfying an $I_{\log}$ condition.