The infinite Brownian loop on a symmetric space

Abstract

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 $X_t\in {\bar{\mathfrak{a}}}_{+}$, 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 $\{X_{tT}/\sqrt{T},t\ge 0\}$ 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 $D=\mathrm{rank}\mathbb{M}+2j$, where $j$ denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on ${\Phi }_0$.