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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 $B_N, C_N, D_N$.

We study Jacobi processes $(X_{t})_{t\ge0}$ on the compact spaces $[-1,1]^N$ and on the noncompact spaces $[1,\infty[^N$ which are motivated by the Heckman-Opdam theory for the root systems of type BC and associated integrable particle systems. These processes depend on three positive parameters and degenerate in the freezing limit to solutions of deterministic dynamical systems. In the compact case, these models tend for $t\to\infty$ to the distributions of the $\beta$-Jacobi ensembles and, in the freezing case, to vectors consisting of ordered zeros of one-dimensional Jacobi polynomials. Representing these processes by stochastic differential equations, we derive almost sure analogues of Wigner's semicircle and Marchenko-Pastur limit laws for $N\to\infty$ for the empirical distributions of the $N$ particles on some local scale. We there allow for arbitrary initial conditions, which enter the limiting distributions via free convolutions These results generalize corresponding stationary limit results in the compact case for $\beta$-Jacobi ensembles and, in the deterministic case, for the empirical distributions of the ordered zeros of Jacobi polynomials by Dette and Studden. The results are also related to free limit theorems for multivariate Bessel processes, $\beta$-Hermite and $\beta$-Laguerre ensembles, and the asymptotic empirical distributions of the zeros of Hermite and Laguerre polynomials for $N\to\infty$.