Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### The infinite Brownian loop on a symmetric space

Auteur(s):

Code(s) de Classification MSC:

• 43A85 Analysis on homogeneous spaces
Résumé: The infinite Brownian loop $\{B_t^0,t\ge 0\}$ on a Riemannian manifold~$\M$ is the limit in distribution of the Brownian bridge of length~$T$ around a fixed origin~$0$, when $T\to+\infty$. 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 $\Phi_0$--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 $D=\rank\M+2j$, where $j$ denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on~$\Phi_0$.