Isomorphisms of $β$-Dyson's Brownian motion with Brownian local time

Abstract: We show that the Brydges-Fr\"ohlich-Spencer-Dynkin and the Le Jan's isomorphisms between the Gaussian free fields and the occupation times of symmetric Markov processes generalize to the $\beta$-Dyson's Brownian motion. For $\beta\in{1,2,4}$ this is a consequence of the Gaussian case, however the relation holds for general $\beta$. We further raise the question whether there is an analogue of $\beta$-Dyson's Brownian motion on general electrical networks, interpolating and extrapolating the fields of eigenvalues in matrix-valued Gaussian free fields. In the case $n=2$ we give a simple construction.