Probability Theory and Related Fields

, Volume 166, Issue 3–4, pp 659–712 | Cite as

Limits of random differential equations on manifolds

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Abstract

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 \(\mathrm{ker}({\mathcal {L}}_0^*)\). Under Hörmander 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.

Mathematics Subject Classification

60H 60J 60F 60D 

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Mathematics InstituteThe University of WarwickCoventryUK

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