Eigenvalue processes of symmetric tridiagonal matrix-valued processes associated with Gaussian beta ensemble

Satoshi Yabuoku

Stochastic Processes and their Applications, 2023, vol. 164, issue C, 139-159

Abstract: We consider the symmetric tridiagonal matrix-valued process associated with Gaussian beta ensemble (GβE) by putting independent Brownian motions and Bessel processes on the diagonal entries and upper (lower)-diagonal ones, respectively. Then, we derive the stochastic differential equations that the eigenvalue processes satisfy, and we show that eigenvalues of their (indexed) principal minor sub-matrices appear in the stochastic differential equations. By the Cauchy’s interlacing argument for eigenvalues, we can characterize the sufficient condition that the eigenvalue processes never collide with each other almost surely, by the dimensions of the Bessel processes.

Keywords: Gaussian beta ensemble; Dyson’s Brownian motion; Eigenvalue processes; Minor eigenvalues; Non-colliding processes (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:164:y:2023:i:c:p:139-159

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DOI: 10.1016/j.spa.2023.07.004

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