The infinite Brownian loop on a symmetric space

  • Jean-Philippe Anker

    CNRS-Université d'Orléans, Orléans, France
  • Philippe Bougerol

    Université Paris 6, Paris, France
  • Thierry Jeulin

    Université Paris 7, Paris, France


The infinite Brownian loop {Bt0,t0}\{B_t^0,t\ge 0\} on a Riemannian manifold M\mathbb{M} is the limit in distribution of the Brownian bridge of length TT around a fixed origin 00, when T+T\to+\infty. It has no spectral gap. When M\mathbb{M} has nonnegative Ricci curvature, B0B^0 is the Brownian motion itself. When M=G/K\mathbb{M}=G/K is a noncompact symmetric space, B0B^0 is the relativized Φ0\Phi_0-process of the Brownian motion, where Φ0\Phi_0 denotes the basic spherical function of Harish-Chandra, i.e. the KK-invariant ground state of the Laplacian. In this case, we consider the polar decomposition Bt0=(Kt,Xt)B_t^0=(K_t,X_t), where KtK/MK_t\in K/M and Xt\conecX_t\in\conec, the positive Weyl chamber. Then, as t+t\to+\infty, KtK_t converges and d(0,Xt)/t0d(0,X_t)/t\to0 almost surely. Moreover the processes {XtT/T,t0}\{X_{tT}/\sqrt{T},t\ge 0\} converge in distribution, as T+T\to+\infty, to the intrinsic Brownian motion of the Weyl chamber. This implies in particular that d(0,XtT)/Td(0,X_{tT})/\sqrt{T} converges to a Bessel process of dimension D=rankM+2jD=rank \mathbb{M}+2j, where jj denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on Φ0\Phi_0.

Cite this article

Jean-Philippe Anker, Philippe Bougerol, Thierry Jeulin, The infinite Brownian loop on a symmetric space. Rev. Mat. Iberoam. 18 (2002), no. 1, pp. 41–97

DOI 10.4171/RMI/311