Covariation representations for Hermitian L\'{e}vy process ensembles of free infinitely divisible distributions

TL;DR

This paper explores Hermitian Le9vy processes linked to free infinitely divisible distributions via matrix ensembles, providing path approximations and showing that rank-one jump matrix subordinators are quadratic variations of complex Le9vy processes.

Contribution

It introduces a new connection between matrix subordinators with rank-one jumps and complex Le9vy processes, expanding understanding of free probability and matrix process representations.

Findings

Path approximation by covariation processes for matrix Le9vy processes.

Any rank-one jump matrix subordinator is the quadratic variation of a complex Le9vy process.

Results apply to matrix ensembles associated with 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 $(M_{d})_{d\geq1}$ whose asymptotic spectral distributions are free infinitely divisible. We investigate Hermitian L\'{e}vy processes with jumps of rank one associated to these random matrix ensembles introduced in [6] and [10]. A sample path approximation by covariation processes for these matrix L\'{e}vy processes is obtained. As a general result we prove that any $d\times d$ complex matrix subordinator with jumps of rank one is the quadratic variation of an $\mathbb{C}^{d}$-valued L\'{e}vy process. In particular, we have the corresponding result for matrix subordinators with jumps of rank one associated to the random matrix ensembles $(M_{d})_{d\geq1}$