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Summary: The in nite Brownian loop on a symmetric space
Jean{Philippe Anker z
Universite Nancy 1
Philippe Bougerol x
Universite Paris 6
Thierry Jeulin {
Universite Paris 7
To appear in Rev. Mat. Iberoamericana 18 (2002), 41{97
Abstract
The in nite Brownian loop fB 0
t
; t 0g on a Riemannian manifold M is
the limit in distribution of the Brownian bridge of length T around a xed
origin 0, when T ! +1. It has no spectral gap. When M has nonnega-
tive Ricci curvature, B 0 is the Brownian motion itself. When M = G=K
is a noncompact symmetric space, B 0 is the relativized 0 {process of the
Brownian motion, where 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 0
t = (K t ; X t ), where K t 2 K=M and
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