On the process of the eigenvalues of a Hermitian L\'evy process

TL;DR

This paper describes the stochastic dynamics of eigenvalues of Hermitian Levy processes, generalizing Dyson-Brownian motion, and studies the conditions under which eigenvalues jump simultaneously based on matrix commutativity and jump rank.

Contribution

It extends the understanding of eigenvalue dynamics for Hermitian Levy processes, including jump behavior and conditions for simultaneous eigenvalue jumps.

Findings

Eigenvalues follow Itô SDEs with jumps.

Simultaneous jumps depend on matrix commutativity.

Full rank jumps cause all eigenvalues to jump in the commutative case.

Abstract

The dynamics of the eigenvalues (semimartingales) of a L\'{e}vy process $X$ with values in Hermitian matrices is described in terms of It\^{o} stochastic differential equations with jumps. This generalizes the well known Dyson-Brownian motion. The simultaneity of the jumps of the eigenvalues of $X$ is also studied. If $X$ has a jump at time $t$ two different situations are considered, depending on the commutativity of $X(t)$ and $X(t-)$. In the commutative case all the eigenvalues jump at time $t$ only when the jump of $X$ is of full rank. In the noncommutative case, $X$ jumps at time $t$ if and only if all the eigenvalues jump at that time when the jump of $X$ is of rank one.