Asymptotics for the infinite Brownian loop on noncompact symmetric spaces

TL;DR

This paper investigates the long-time behavior of the infinite Brownian loop on noncompact symmetric spaces, revealing convergence properties similar to Euclidean spaces and extending understanding of heat equation solutions under Doob transform.

Contribution

It provides the first analysis of the infinite Brownian loop's asymptotics on noncompact symmetric spaces, highlighting convergence phenomena without symmetry restrictions.

Findings

L^1 asymptotic convergence without bi-K-invariance

Strong L^∞ convergence of solutions

Behavior analogous to Euclidean setting

Abstract

The infinite Brownian loop on a Riemannian manifold is the limit in distribution of the Brownian bridge of length $T$ around a fixed origin when $T \rightarrow +\infty$. The aim of this note is to study its long-time asymptotics on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. This amounts to the behavior of solutions to the heat equation subject to the Doob transform induced by the ground spherical function. Unlike the standard Brownian motion, we observe in this case phenomena which are similar to the Euclidean setting, namely $L^1$ asymptotic convergence without requiring bi-$K$-invariance for initial data, and strong $L^{\infty}$ convergence.