Random perturbation to the geodesic equation


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 11. We prove that the projections of the solutions to the perturbed equations, converge, after suitable rescaling, to a Brownian motion scaled by 8n(n1){\frac{8}{n(n-1)}} where nn is the dimension of the state space. Their horizontal lifts to the orthonormal frame bundle converge also, to a scaled horizontal Brownian motion.Comment: Published at in the Annals of Probability ( by the Institute of Mathematical Statistics (

    Similar works