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 ∑ k Ykαk(z t (ω)) where Yk are vector fields, is a positive number, z t is a 1 L0 diffusion process taking values in possibly a different manifold, αk are annihilators of ker(L0). Under Hörmander type conditions on L0 we prove that, as approaches zero, the stochastic processes y t converge weakly and in the Wasserstein topologies. We describe this limit and give an upper bound for the rate of the convergence.