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European Mathematical Society Publishing House
2016-09-19 17:05:43
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
18
2002
1
The infinite Brownian loop on a symmetric space
Jean-Philippe
Anker
CNRS-Université d'Orléans, ORLÉANS CEDEX 2, FRANCE
Philippe
Bougerol
Université Paris 6, PARIS CEDEX 05, FRANCE
Thierry
Jeulin
Université Paris 7, PARIS CEDEX 05, FRANCE
Brownian bridge, central limit theorem, ground state, heat kernel, quotient limit theorem, relativized process, Riemannian manifold, spherical function, symmetric space, Weyl chamber
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 $\Phi_0$.
Abstract harmonic analysis
Differential geometry
Global analysis, analysis on manifolds
Probability theory and stochastic processes
41
97
10.4171/RMI/311
http://www.ems-ph.org/doi/10.4171/RMI/311