The infinite Brownian loop on a symmetric space

The infinite Brownian loop \{B_t^0,t≥ 0\} on a Riemannian manifold M is the limit in distribution of the Brownian bridge of length T around a fixed origin 0 , when T→+∞ . It has no spectral gap. When M has nonnegative Ricci curvature, B^0 is the Brownian motion itself. When M=G/K is a noncompact symmetric space, B^0 is the relativized Φ_0 -process of the Brownian motion, where Φ_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∈ K/M and X_t∈ ā_+ , the positive Weyl chamber. Then, as t→+∞ , K_t converges and d(0,X_t)/t→ 0 almost surely. Moreover the processes \{XtT/√T,t≥ 0\} converge in distribution, as T→+∞ , to the intrinsic Brownian motion of the Weyl chamber. This implies in particular that d(0,XtT)/√T converges to a Bessel process of dimension D=rank M+2j , where j denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on Φ_0 .