64,211 research outputs found

    Random perturbation to the geodesic equation

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    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 11. We prove that the projections of the solutions to the perturbed equations, converge, after suitable rescaling, to a Brownian motion scaled by 8n(nβˆ’1){\frac{8}{n(n-1)}} where nn is the dimension of the state space. Their horizontal lifts to the orthonormal frame bundle converge also, to a scaled horizontal Brownian motion.Comment: Published at http://dx.doi.org/10.1214/14-AOP981 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    On the Semi-Classical Brownian Bridge Measure

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    We prove an integration by parts formula for the probability measure induced by the semi-classical Riemmanian Brownian bridge over a manifold with a pole

    Limits of Random Differential Equations on Manifolds

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    Consider a family of random ordinary differential equations on a manifold driven by vector fields of the form βˆ‘kYkΞ±k(ztΟ΅(Ο‰))\sum_kY_k\alpha_k(z_t^\epsilon(\omega)) where YkY_k are vector fields, Ο΅\epsilon is a positive number, ztΟ΅z_t^\epsilon is a 1Ο΅L0{1\over \epsilon} {\mathcal L}_0 diffusion process taking values in possibly a different manifold, Ξ±k\alpha_k are annihilators of ker(L0βˆ—)ker ({\mathcal L}_0^*). Under H\"ormander type conditions on L0{\mathcal L}_0 we prove that, as Ο΅\epsilon approaches zero, the stochastic processes ytϡϡy_{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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