Kinetic Dyson Brownian motion. (English) Zbl 1498.60331

The author studies kinetic Dyson Brownian motions as follows: Let \((H_t^.)_{t\ge0}\) be a Brownian motion on the unit sphere in the vector space of all \(d\times d\) Hermitian matrices, and \((H_t=\int_0^t H_s^.ds)_{t\ge 0}\) the associated velocity spherical Brownian motion. The \(2d\)-dimensional processes \((L_t,L_t^. )_{t\ge0}\) of the the associated eigenvalues then are called kinetic Dyson Brownian motions.
The author shows that these \(2d\)-dimensional processes are Markovian precisely for \(d=2\), and that \((L_t )_{t\ge0}\) tends to the usual Dyson Brownian motion when taking some canonical space-time-scalings and limit.


60J65 Brownian motion
60B20 Random matrices (probabilistic aspects)
60G53 Feller processes
Full Text: DOI arXiv


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