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Invariance Principle for Lifts of Geodesic Random Walks.

Jonathan Junné1, Frank Redig1, Rik Versendaal1

  • 1Delft Institute of Applied Mathematics, TU Delft, Mekelweg 4, 2628 CD Delft, Netherlands.

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|February 26, 2026
PubMed
Summary

Researchers studied geodesic random walks on Riemannian submersions, proving convergence to horizontal Brownian motion. This provides a probabilistic proof for a geometric identity linking Laplacians on different manifolds.

Keywords:
Geodesic random walksHorizontal LaplacianInvariance principleRiemannian Brownian motionRiemannian submersion

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Area of Science:

  • Differential Geometry
  • Stochastic Analysis
  • Probability Theory

Background:

  • Riemannian submersions are fundamental geometric structures.
  • Geodesic random walks are essential tools in analyzing manifolds.
  • Understanding the relationship between different operators on manifolds is crucial.

Purpose of the Study:

  • To investigate lifted geodesic random walks on Riemannian submersions.
  • To establish an invariance principle for these walks.
  • To provide a probabilistic proof of a geometric identity involving Laplacians.

Main Methods:

  • Analysis of Riemannian submersions.
  • Study of geodesic random walks and their lifted counterparts.
  • Application of invariance principles and convergence theorems.
  • Probabilistic methods to prove geometric identities.

Main Results:

  • An invariance principle is proven for lifted geodesic random walks under specific conditions.
  • Convergence to horizontal Brownian motion is established for these walks.
  • A natural probabilistic proof of the geometric identity relating horizontal and Laplace-Beltrami operators is presented.

Conclusions:

  • The study establishes a significant link between stochastic processes and geometric structures.
  • The findings offer a new perspective on proving geometric identities.
  • The results are relevant to the construction of Riemannian Brownian motion, particularly in the context of orthonormal frame bundles.