  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.
