May 2023 On the singular values of complex matrix Brownian motion with a matrix drift
Theodoros Assiotis
Author Affiliations +
Bernoulli 29(2): 1686-1713 (May 2023). DOI: 10.3150/22-BEJ1517


Let MatC(K,N) be the space of K×N complex matrices. Let Bt be Brownian motion on MatC(K,N) starting from the zero matrix and MMatC(K,N). We prove that, with KN, the N eigenvalues of Bt+tMBt+tM form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman (Ann. Probab. 9 (1981) 573–582) for multidimensional Brownian motion with drift which corresponds to N=1. We then give two more descriptions for this Markov process. First, as independent squared Bessel diffusion processes in the wide sense, introduced by Watanabe (Z. Wahrsch. Verw. Gebiete 31 (1975) 115–124) and studied by Pitman and Yor (In Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980) (1981) 285–370 Springer), conditioned to never intersect. Second, as the distribution of the top row of interacting squared Bessel type diffusions in some interlacting array. The last two descriptions also extend to a general class of one-dimensional diffusions.


I am very grateful to Jon Warren and Neil O’Connell for useful discussions. I am also very grateful to two anonymous referees for a careful reading of the paper and many useful comments and suggestions which have improved the paper.


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Theodoros Assiotis. "On the singular values of complex matrix Brownian motion with a matrix drift." Bernoulli 29 (2) 1686 - 1713, May 2023.


Received: 1 July 2021; Published: May 2023
First available in Project Euclid: 19 February 2023

MathSciNet: MR4550241
zbMATH: 1510.60072
Digital Object Identifier: 10.3150/22-BEJ1517

Keywords: Interacting diffusions , interlacing arrays , matrix Brownian motion with drift , Non-intersecting paths , singular values , squared Bessel diffusions in wide sense


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Vol.29 • No. 2 • May 2023
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