3,727 research outputs found

### Random perturbation to the geodesic equation

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 $1$. We
prove that the projections of the solutions to the perturbed equations,
converge, after suitable rescaling, to a Brownian motion scaled by
${\frac{8}{n(n-1)}}$ where $n$ 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

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 $ker ({\mathcal
L}_0^*)$. Under H\"ormander 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.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

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

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

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 $t$, the solution flow $F_t$ is
weakly differentiable and for each $p>0$ there is a positive number $T(p)$ such
that for all $t<T(p)$, the solution flow $F_t(\cdot)$ 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

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