Invariance principle for Lifts of Geodesic Random Walks

2023/07/05 by Junné, Jonathan, Redig, Frank, Versendaal, Rik
#58J65 #60J65 #60K35 #Differential Geometry (math.DG) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2307.02160

Abstract

We consider a certain class of Riemannian submersions 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 Δ_\H on N and the Laplace-Beltrami operator Δ_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.

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