Kinetic Dyson Brownian motion

Abstract

We study the spectrum of the kinetic Brownian motion in the space of $d\times d$ Hermitian matrices, $d\ge 2$. We show that the eigenvalues stay distinct for all times, and that the process Λ of eigenvalues is a kinetic diffusion (i.e. the pair $(\mathrm{\Lambda },\dot{\mathrm{\Lambda }})$ of Λ and its derivative is Markovian) if and only if $d=2$. In the large scale and large time limit, we show that Λ converges to the usual (Markovian) Dyson Brownian motion under suitable normalisation, regardless of the dimension.