## The Annals of Probability

### Random perturbation to the geodesic equation

Xue-Mei Li

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.aop/1454423049

Digital Object Identifier
doi:10.1214/14-AOP981

Mathematical Reviews number (MathSciNet)
MR3456345

Zentralblatt MATH identifier
1372.60083

#### Citation

Li, Xue-Mei. Random perturbation to the geodesic equation. Ann. Probab. 44 (2016), no. 1, 544--566. doi:10.1214/14-AOP981. https://projecteuclid.org/euclid.aop/1454423049

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