Limits of Random Differential Equations on Manifolds

Consider a family of random ordinary differential equations on a manifold driven by vector fields of the form $\sum_kY_k\alpha_k(z_t^\epsilon(\omega))$ where $Y_k$ are vector fields, $\epsilon$ is a positive number, $z_t^\epsilon$ is a ${1\over \epsilon} {\mathcal L}_0$ diffusion process taking values in possibly a different manifold, $\alpha_k$ are annihilators of $ker ({\mathcal L}_0^*)$. Under H\"ormander type conditions on ${\mathcal L}_0$ we prove that, as $\epsilon $ approaches zero, the stochastic processes $y_{t\over \epsilon}^\epsilon$ converge weakly and in the Wasserstein topologies. We describe this limit and give an upper bound for the rate of the convergence.