An explicit formula for free multiplicative Brownian motions via spherical functions

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 $\mathbb R^N$ of type $A_{N-1}$. 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 $N\to\infty$ 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 $B_N, C_N, D_N$.

💡 Research Summary

The paper investigates the relationship between Brownian motions on the general linear groups GL(N,F) (F = ℝ, ℂ) and certain integrable particle systems known as Heckman‑Opdam processes. After normalising, the logarithms of the ordered singular values of a right‑Brownian motion Gₜ on GL(N,F) evolve inside the Weyl chamber C_Aₙ and satisfy the stochastic differential equations

 dXₜ,i = dBₜ,i + k ∑_{j≠i}coth(Xₜ,i−Xₜ,j) dt, i=1,…,N,

where k>0 is a coupling constant (k=½ corresponds to the real case, k=1 to the complex case). For k≥½ these SDEs have unique strong solutions and, in the limit k→∞, they degenerate to a deterministic ODE.

The main probabilistic object of interest is the empirical measure

 μ_{N,t}= (1/N)∑{i=1}^N δ{Xₜ,i/N}.

The authors prove that for every k∈