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Limits of Random Differential Equations on Manifolds

Abstract

Consider a family of random ordinary differential equations on a manifold driven by vector fields of the form kYkαk(ztϵ(ω))\sum_kY_k\alpha_k(z_t^\epsilon(\omega)) where YkY_k are vector fields, ϵ\epsilon is a positive number, ztϵz_t^\epsilon is a 1ϵL0{1\over \epsilon} {\mathcal L}_0 diffusion process taking values in possibly a different manifold, αk\alpha_k are annihilators of ker(L0)ker ({\mathcal L}_0^*). Under H\"ormander type conditions on L0{\mathcal L}_0 we prove that, as ϵ\epsilon approaches zero, the stochastic processes ytϵϵy_{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.Comment: 46 pages, To appear in Probability Theory and Related Fields In this version, we add a note in proof for the published versio

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