The infinite Brownian loop $\{B_t^0,t\ge 0\}$ on a Riemannian
manifold $\mathbb{M}$ is the limit in distribution of the Brownian bridge of
length $T$ around a fixed origin $0$, when $T\to+\infty$. It has no
spectral gap. When $\mathbb{M}$ has nonnegative Ricci curvature, $B^0$ is the
Brownian motion itself. When $\mathbb{M}=G/K$ is a noncompact symmetric
space, $B^0$ is the relativized $\Phi_0$-process of the Brownian
motion, where $\Phi_0$ denotes the basic spherical function of
Harish-Chandra, i.e. the $K$-invariant ground state of the
Laplacian. In this case, we consider the polar decomposition
$B_t^0=(K_t,X_t)$, where $K_t\in K/M$ and $X_t\in\conec$, the
positive Weyl chamber. Then, as $t\to+\infty$, $K_t$ converges and
$d(0,X_t)/t\to0$ almost surely. Moreover the processes
$\{X_{tT}/\sqrt{T},t\ge 0\}$ converge in distribution, as
$T\to+\infty$, to the intrinsic Brownian motion of the Weyl chamber.
This implies in particular that $d(0,X_{tT})/\sqrt{T}$ converges to a
Bessel process of dimension $D=rank \mathbb{M}+2j$, where $j$ denotes the
number of positive indivisible roots. An ingredient of the proof is a
new estimate on $\Phi_0$.