Covariation representations for Hermitian Lévy process ensembles of free infinitely divisible distributions

Abstract

It is known that the so-called Bercovici-Pata bijection can be explained in terms of certain Hermitian random matrix ensembles (M_d)_d≥1 whose asymptotic spectral distributions are free infinitely divisible. We investigate Hermitian Lévy processes with jumps of rank one associated to these random matrix ensembles introduced by Benaych-Georges and Cabanal-Duvillard. A sample path approximation by covariation processes for these matrix Lévy processes is obtained. As a general result we prove that any d×d complex matrix subordinator with jumps of rank one is the quadratic variation of a $\mathbb{C}^d$-valued Lévy process. In particular, we have the corresponding result for matrix subordinators with jumps of rank one associated to the random matrix ensembles (M_d)_d≥1.