  Multivariate Bessel processes $(X_{t,k})_{t\ge0}$ describe interacting
particle systems of Calogero-Moser-Sutherland type and are related with
$\beta$-Hermite and $\beta$-Laguerre ensembles. They depend on a root system
and a multiplicity $k$ which corresponds to the parameter $\beta$ in random
matrix theory. In the recent years, several limit theorems were derived for
$k\to\infty$ with fixed $t>0$ and fixed starting point. Only recently, Andraus
and Voit used the stochastic differential equations of $(X_{t,k})_{t\ge0}$ to
derive limit theorems for $k\to\infty$ with starting points of the form $\sqrt
k\cdot x$ with $x$ in the interior of the corresponding Weyl chambers. Here we
provide associated functional central limit theorems which are locally uniform
in $t$. The Gaussian limiting processes admit explicit representations in terms
of matrix exponentials and the solutions of the associated deterministic
dynamical systems.
