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

Wallach sets and squared Bessel particle systems