Revista Matemática Iberoamericana

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Volume 18, Issue 1, 2002, pp. 41–97
DOI: 10.4171/RMI/311

Published online: 2002-04-30

The infinite Brownian loop on a symmetric space

Jean-Philippe Anker[1], Philippe Bougerol[2] and Thierry Jeulin[3]

(1) CNRS-Université d'Orléans, Orléans, France
(2) Université Paris 6, Paris, France
(3) Université Paris 7, Paris, France

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 $\Phi_0$.

Keywords: Brownian bridge, central limit theorem, ground state, heat kernel, quotient limit theorem, relativized process, Riemannian manifold, spherical function, symmetric space, Weyl chamber

Anker Jean-Philippe, Bougerol Philippe, Jeulin Thierry: The infinite Brownian loop on a symmetric space. Rev. Mat. Iberoam. 18 (2002), 41-97. doi: 10.4171/RMI/311