A new approach for the unitary Dyson Brownian motion through the theory of viscosity solutions

In this paper, we study the unitary Dyson Brownian motion through a partial differential equation approach recently introduced for the real Dyson case. The main difference with the real Dyson case is that the spectrum is now on the circle and not on the real line, which leads to particular attention to comparison principles. First we recall why the system of particles which are the eigenvalues of unitary Dyson Brownian motion is well posed thanks to a containment function. Then we proved that the primitive of the limit spectral measure of the unitary Dyson Brownian motion is the unique solution to a viscosity equation obtained by primitive the Dyson equation on the circle. Finally, we study some properties of solutions of Dyson’s equation on the circle. We prove a L $\infty$ regularization. We also look at the long time behaviour in law of a solution through a study of the so-called free entropy of the system. We conclude by discussing the uniform convergence towards the uniform measure on the circle of a solution of the Dyson equation.

💡 Research Summary

The paper investigates the unitary Dyson Brownian motion (UDBM), a stochastic dynamics of eigen‑angles of large random unitary matrices, by recasting it as a nonlinear partial differential equation (PDE) on the unit circle and applying the theory of viscosity solutions.

Model and well‑posedness.
For an N×N matrix drawn from the Gaussian Unitary Ensemble, the eigenvalues can be written as e^{iθ_j(t)}. Dyson’s original stochastic differential system reads
dθ_i = (1/N)∑_{j≠i} cot