An explicit formula for free multiplicative Brownian motions via spherical functions
Martin Auer, Michael Voit
TL;DR
The paper connects Brownian motions on GL(N, F) to Weyl-group invariant Heckman-Opdam processes via the logarithms of ordered singular values, and uses spherical functions of complex Cartan motion groups to reinterpret complex-case dynamics as drifted Brownian motions on Hermitian matrices. It derives explicit density and generator formulas for the projected processes, enabling a rigorous large-N analysis that reveals a k-independent free-limit for the empirical measures and identifies the limit with free multiplicative convolution, including a concrete realization in terms of Biane's free multiplicative Brownian motion. The results extend beyond type A to root systems B_N, C_N, D_N, providing analogous explicit densities and free-limit descriptions, thereby linking stochastic processes, harmonic analysis, and free probability in a unified framework.
Abstract
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 $B_N, C_N, D_N$.