Open Access
2022 Kinetic Dyson Brownian motion
Pierre Perruchaud
Author Affiliations +
Electron. Commun. Probab. 27: 1-12 (2022). DOI: 10.1214/22-ECP480


We study the spectrum of the kinetic Brownian motion in the space of d×d Hermitian matrices, d2. 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 (Λ,Λ˙) 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.


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.


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Pierre Perruchaud. "Kinetic Dyson Brownian motion." Electron. Commun. Probab. 27 1 - 12, 2022.


Received: 27 January 2022; Accepted: 25 July 2022; Published: 2022
First available in Project Euclid: 11 August 2022

arXiv: 2101.10426
MathSciNet: MR4478123
zbMATH: 1498.60331
Digital Object Identifier: 10.1214/22-ECP480

Primary: 60B20 , 60G53 , 60J60

Keywords: Markov process , random matrices , Stochastic differential equations

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