The infinite Brownian loop on a symmetric space.
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 \(\Phi_0\).
scientific article; zbMATH DE number 1803983
scientific article