# Invariance Principle for Lifts of Geodesic Random Walks

## TL;DR

This paper proves a probabilistic invariance principle for lifted geodesic random walks on manifolds, linking horizontal Brownian motion to geometric identities.

## Contribution

A new probabilistic proof of a geometric identity involving horizontal Laplacians and Laplace–Beltrami operators is established.

## Key findings

- Lifted geodesic random walks converge to horizontal Brownian motion under specific speed distribution conditions.
- The invariance principle provides a natural proof of the identity between horizontal Laplacians and base manifold Laplace–Beltrami operators.
- The result is significant for constructing Riemannian Brownian motion in the orthonormal frame bundle setting.

## Abstract

We consider a certain class of Riemannian submersions \documentclass[12pt]{minimal}
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\begin{document}$$\pi : N \rightarrow M$$\end{document}π:N→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 \documentclass[12pt]{minimal}
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\begin{document}$$\Delta _{\mathcal {H}}$$\end{document}ΔH on N and the Laplace–Beltrami operator \documentclass[12pt]{minimal}
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\begin{document}$$\Delta _M$$\end{document}ΔM on M. In the setting where N is the orthonormal frame bundle O(M), this identity is central in the Malliavin–Eells–Elworthy construction of Riemannian Brownian motion.