Invariance principle for Lifts of Geodesic Random Walks

TL;DR

This paper proves an invariance principle for lifted geodesic random walks on Riemannian submersions, showing convergence to horizontal Brownian motion and providing a probabilistic proof of a key geometric Laplacian identity.

Contribution

It establishes an invariance principle for geodesic random walks under Riemannian submersions, linking probabilistic convergence to geometric Laplacian identities.

Findings

Convergence of lifted geodesic random walks to horizontal Brownian motion.

Probabilistic proof of the identity between horizontal Laplacian and Laplace-Beltrami operator.

Application to the orthonormal frame bundle and Brownian motion construction.

Abstract

We consider a certain class of Riemannian submersions $\pi : N \to M$ and study lifted geodesic random walks from the base manifold $M$ to the total manifold $N$. Under appropriate conditions on the distribution of the speed of the geodesic random walks, we prove an invariance principle; i.e., convergence to horizontal Brownian motion for the lifted walks. This gives us a natural probabilistic proof of the geometric identity relating the horizontal Laplacian $\Delta_\H$ on $N$ and the Laplace-Beltrami operator $\Delta_M$ on $M$. In particular, when $N$ is the orthonormal frame bundle $O(M)$, this identity is central in the Malliavin-Eells-Elworthy construction of Riemannian Brownian motion.