Wallach sets and squared Bessel particle systems

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 $\overline{Sy{m}^{+}(\mathbf{R},p)}$. 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.