Recent advances on eigenvalues of matrix-valued stochastic processes

TL;DR

This survey reviews three decades of research on the eigenvalues of matrix-valued stochastic processes, including recent developments involving fractional Brownian motion and Brownian sheets, highlighting key results and open problems.

Contribution

It provides a comprehensive overview of the evolution and recent advances in understanding eigenvalues of matrix-valued stochastic processes over the past thirty years.

Findings

Analysis of eigenvalue behaviors in high-dimensional limits

Discussion of eigenvalues driven by fractional Brownian motion

Identification of open problems in the field

Abstract

Since the introduction of Dyson's Brownian motion in early 1960's, there have been a lot of developments in the investigation of stochastic processes on the space of Hermitian matrices. Their properties, especially, the properties of their eigenvalues have been studied in great details. In particular, the limiting behaviors of the eigenvalues are found when the dimension of the matrix space tends to infinity, which connects with random matrix theory. This survey reviews a selection of results on the eigenvalues of stochastic processes from the literature of the past three decades. For most recent variations of such processes, such as matrix-valued processes driven by fractional Brownian motion or Brownian sheet, the eigenvalues of them are also discussed in this survey. In the end, some open problems in the area are also proposed.