The infinite Brownian loop on a symmetric space.
type: Item
instance of: scholarly article
title: The infinite Brownian loop on a symmetric space.
zbMATH Open document ID: 1090.58020
DOI: 10.4171/RMI/311
author: Jean-Philippe Anker
author: Philippe Bougerol
author: Thierry Jeulin
published in: Revista Matemática Iberoamericana
publication date: 2002-11-24T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmi/1045578693
full work available at URL: https://eudml.org/doc/39647
review text: Summary: The infinite Brownian loop \(\{B_t^0,t\geq 0\}\) 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 \(X_t\in\overline{\mathfrak a}_+\), the positive Weyl chamber. Then, as \(t\to+ \infty\), \(K_t\) converges and \(d(0,X_t)/t\to 0\) almost surely. Moreover, the processes \(\{X_{tT}/\sqrt T,t\geq 0\}\) 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 \(D=\text{rank}\,\mathbb{M}+2j\), where \(j\) denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on \(\Phi_0\).
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 1803983
zbMATH Keywords: Brownian bridge
zbMATH Keywords: central limit theorem
zbMATH Keywords: ground state
zbMATH Keywords: heat kernel
zbMATH Keywords: quotient limit theorem
zbMATH Keywords: relativized process
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: spherical function
zbMATH Keywords: symmetric space
zbMATH Keywords: Weyl chamber
MaRDI profile type: Publication
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The infinite Brownian loop on a symmetric space. (Wikidata QID)
Wikidata QID: Q125351253