Wallach sets and squared Bessel particle systems

TL;DR

This paper characterizes the classical and non-central Wallach sets using probabilistic methods, proving the Mayerhofer conjecture by linking Wallach sets to squared Bessel matrix processes and their eigenvalues.

Contribution

It provides a probabilistic characterization of Wallach sets and proves the Mayerhofer conjecture using SDE analysis of squared Bessel processes and eigenvalue dynamics.

Findings

Complete description of Wallach sets $W_0$ and $W$

Proof of the Mayerhofer conjecture

Connection between Wallach sets and squared Bessel matrix processes

Abstract

We determine the classical and the non-central Wallach sets $W_0$ and $W$ by classical probabilistic methods. We prove the Mayerhofer conjecture on $W$. We exploit the fact that $(x_0,\beta)\in W$ if and only if $x_0$ is the starting point and $2\beta$ is the drift of a squared Bessel matrix process $X_t$ on the cone $\bar{Sym^+(\mathbf{R},p)}$. Our methods are based on the study of SDEs for the symmetric polynomials of $X_t$ and for the eigenvalues of $X_t$, i.e. the squared Bessel particle systems.