3,727 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(n1){\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

    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 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

    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

    First Order Feynman-Kac Formula

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    We study the parabolic integral kernel associated with the weighted Laplacian and the Feynman-Kac kernels. For manifold with a pole we deduce formulas and estimates for them and for their derivatives, given in terms of a Gaussian term and the semi-classical bridge. Assumptions are on the Riemannian data.Comment: 31 pages, to appear in `Stochastic Processes and their Applications

    Strong completeness for a class of stochastic differential equations with irregular coefficients

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    We prove the strong completeness for a class of non-degenerate SDEs, whose coefficients are not necessarily uniformly elliptic nor locally Lipschitz continuous nor bounded. Moreover, for each tt, the solution flow FtF_t is weakly differentiable and for each p>0p>0 there is a positive number T(p)T(p) such that for all t<T(p)t<T(p), the solution flow Ft()F_t(\cdot) belongs to the Sobolev space W_{\loc}^{1,p}. The main tool for this is the approximation of the associated derivative flow equations. As an application a differential formula is also obtained