2021/07/11 by Theodoros Assiotis, Assiotis, Theodoros
Mathematics · #Random Matrices and Applications #Advanced Algebra and Geometry #Point processes and geometric inequalities
paper · doi:10.48550/arxiv.2107.05028
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 M∈ Matℂ(K,N). We prove that, with K≥ N, the N eigenvalues of (Bt+tM)^*(Bt+tM) form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman 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 and studied by Pitman and Yor, 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.