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

TL;DR

This paper extends classical isomorphisms linking Gaussian free fields and occupation times to the $eta$-Dyson's Brownian motion for all $eta$, exploring their generalization and potential analogues on electrical networks.

Contribution

It generalizes known isomorphisms to the $eta$-Dyson's Brownian motion for all $eta$, beyond the classical cases of 1, 2, and 4.

Findings

Isomorphisms extend to general $eta$-Dyson's Brownian motion.

For $n=2$, a simple construction of the process is provided.

Raises questions about analogues on electrical networks.

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.