Random perturbation to the geodesic equation

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.

Published at http://dx.doi.org/10.1214/14-AOP981 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• An averaging principle for a completely integrable stochastic Hamiltonian system
• An averaging principle for diffusions in foliated spaces
• Effective Diffusions with Intertwined Structures

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