Abstract
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\bar{\mathfrak a}_+
, the positive Weyl chamber. Then, as
t\to+\infty
,
K_t
converges and
d(0,X_t)/t\to 0
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=\mathrm{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
.