The Annals of Probability

Random perturbation to the geodesic equation

Xue-Mei Li

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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.

Article information

Ann. Probab. Volume 44, Number 1 (2016), 544-566.

Received: August 2014
Revised: October 2014
First available in Project Euclid: 2 February 2016

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 37Hxx: Random dynamical systems [See also 15B52, 34D08, 34F05, 47B80, 70L05, 82C05, 93Exx] 53B05: Linear and affine connections

Horizontal flows horizontal Brownian motions vertical perturbation stochastic differential equations homogenisation geodesics


Li, Xue-Mei. Random perturbation to the geodesic equation. Ann. Probab. 44 (2016), no. 1, 544--566. doi:10.1214/14-AOP981.

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