# The infinite Brownian loop on a symmetric space

### Jean-Philippe Anker

CNRS-Université d'Orléans, Orléans, France### Philippe Bougerol

Université Paris 6, Paris, France### Thierry Jeulin

Université Paris 7, Paris, France

## Abstract

The infinite Brownian loop ${B_{t},t≥0}$ 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 $B_{t}=(K_{t},X_{t})$, where $K_{t}∈K/M$ and $X_{t}∈aˉ_{+}$, the positive Weyl chamber. Then, as $t→+∞$, $K_{t}$ converges and $d(0,X_{t})/t→0$ almost surely. Moreover the processes ${X_{tT}/T ,t≥0}$ converge in distribution, as $T→+∞$, to the intrinsic Brownian motion of the Weyl chamber. This implies in particular that $d(0,X_{tT})/T $ converges to a Bessel process of dimension $D=rankM+2j$, where $j$ denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on $Φ_{0}$.

## Cite this article

Jean-Philippe Anker, Philippe Bougerol, Thierry Jeulin, The infinite Brownian loop on a symmetric space. Rev. Mat. Iberoam. 18 (2002), no. 1, pp. 41–97

DOI 10.4171/RMI/311