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January 2016 Random perturbation to the geodesic equation
Xue-Mei Li
Ann. Probab. 44(1): 544-566 (January 2016). DOI: 10.1214/14-AOP981


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. "Random perturbation to the geodesic equation." Ann. Probab. 44 (1) 544 - 566, January 2016.


Received: 1 August 2014; Revised: 1 October 2014; Published: January 2016
First available in Project Euclid: 2 February 2016

zbMATH: 1372.60083
MathSciNet: MR3456345
Digital Object Identifier: 10.1214/14-AOP981

Primary: 37Hxx , 53B05 , 58J65 , 60H10

Keywords: geodesics , homogenisation , horizontal Brownian motions , Horizontal flows , Stochastic differential equations , vertical perturbation

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 1 • January 2016
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