We study the spectrum of the kinetic Brownian motion in the space of Hermitian matrices, . 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 of Λ and its derivative is Markovian) if and only if . 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.
The author would like to thank Thierry Lévy, who raised the question of existence and behaviour of kinetic Dyson Brownian motion during the former’s PhD defence.
"Kinetic Dyson Brownian motion." Electron. Commun. Probab. 27 1 - 12, 2022. https://doi.org/10.1214/22-ECP480