Dans ce document, 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.

Cet article explore les excursions dans le temps et leurs implications.

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