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arXiv · 1402.5861

Random perturbation to the geodesic equation

Xue-Mei Li
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Abstract

We study random "perturbation" to the geodesic equation. The geodesic equation is identified with a canonical differential equation on the orthonormal frame bundle driven by a horizontal vector field of norm $1$. We prove that the projections of the solutions to the perturbed equations, converge, after suitable rescaling, to a Brownian motion scaled by ${\frac{8}{n(n-1)}}$ where $n$ is the dimension of the state space. Their horizontal lifts to the orthonormal frame bundle converge also, to a scaled horizontal Brownian motion.

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Xue-Mei Li. 2016-02-10. Random perturbation to the geodesic equation. https://doi.org/10.1214/14-aop981

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