×

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.

MSC:

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

References:

[1] J. Angst, I. Bailleul, and P. Perruchaud, Kinetic Brownian motion on the diffeomorphism group of a closed Riemannian manifold, arXiv e-prints (2019), 1905.04103. · Zbl 0071.00414
[2] Jürgen Angst, Ismaël Bailleul, and Camille Tardif, Kinetic Brownian motion on Riemannian manifolds, Electron. J. Probab. 20 (2015), no. 110, 40. · Zbl 1329.60274
[3] V. I. Arnol’d, Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect, Selecta Math. (N.S.) 1 (1995), no. 1, 1-19. · Zbl 0841.58008
[4] Fabrice Baudoin and Camille Tardif, Hypocoercive estimates on foliations and velocity spherical Brownian motion, Kinet. Relat. Models 11 (2018), no. 1, 1-23. · Zbl 1374.60150
[5] Paul Breiding, Khazhgali Kozhasov, and Antonio Lerario, On the geometry of the set of symmetric matrices with repeated eigenvalues, Arnold Math. J. 4 (2018), no. 3-4, 423-443. · Zbl 07146873
[6] Freeman J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J. Mathematical Phys. 3 (1962), 1191-1198. · Zbl 0111.32703
[7] Jacques Franchi, Exact small time equivalent for the density of the circular langevin diffusion, 2015.
[8] Xue-Mei Li, Effective Diffusions with Intertwined Structures, arXiv e-prints (2012), 1204.3250.
[9] Xue-Mei Li, Random perturbation to the geodesic equation, Ann. Probab. 44 (2016), no. 1, 544-566. · Zbl 1372.60083
[10] Aleksandar Mijatović, Veno Mramor, and Gerónimo Uribe Bravo, Projections of spherical Brownian motion, Electron. Commun. Probab. 23 (2018), Paper No. 52, 12. · Zbl 1415.60067
[11] Pierre Perruchaud, Homogenisation for anisotropic kinetic random motions, Electron. J. Probab. 25 (2020), Paper No. 39, 26. · Zbl 1458.58017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.