Kinetic Dyson Brownian motion

We study the spectrum of the kinetic Brownian motion in the space of $d\times d$ Hermitian matrices, $d\geq2$. We show that the eigenvalues stay distinct for all times, and that the process $\Lambda$ of eigenvalues is a kinetic diffusion (i.e. the pair $(\Lambda,\dot\Lambda)$ of $\Lambda$ and its derivative is Markovian) if and only if $d=2$. In the large scale and large time limit, we show that $\Lambda$ converges to the usual (Markovian) Dyson Brownian motion under suitable normalisation, regardless of the dimension.