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 $1$. 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.