2024/08/01 by Martin Auer, Auer, Martin, Michael Voit +1 · 2 citations
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #22E46 #43A62 #43A90 #60B15 #60B20 #60F15 #60J65 #60K35 #70F10 #82C22 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2408.00535
openalex publication_date 2024/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
After some normalization, the logarithms of the ordered singular values of Brownian motions on GL(N,\mathbb F) with \mathbb F=\mathbb R, \mathbb C form Weyl-group invariant Heckman-Opdam processes on \mathbb RN of type AN-1. We use classical elementary formulas for the spherical functions of GL(N,\mathbb C)/SU(N) and the associated Euclidean spaces H(N,\mathbb C) of Hermitian matrices, and show that in the GL(N,\mathbb C)-case, these processes can be also interpreted as ordered eigenvalues of Brownian motions on H(N,\mathbb C) with particular drifts. This leads to an explicit description for the free limits for the associated empirical processes for N→∞ where these limits are independent from the parameter k of the Heckman-Opdam processes. In particular we get new formulas for the distributions of the free multiplicative Browniam motion of Biane. We also show how this approach works for the root systems BN, CN, DN.