75,801 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

### 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

### 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

### Light Curve Properties of Supernovae Associated With Gamma-ray Bursts

Little is known about the diversity in the light curves of GRB-SNe, including
whether the light curve of SN 1998bw can be used as a representative template
or whether there is a luminosity-decline rate relation akin to that of SNe Ia.
In this paper, we aim to obtain well-constrained light curves of GRB-SNe
without the assumption of empirical or parametric templates and to investigate
whether the peak brightness correlates with other parameters such as the light
curve shape or the time of peak. We select eight SNe in the redshift range
0.0085 to 0.606, which are firmly associated with GRBs. The light curves of
these GRB-SNe are well sampled across the peak. Afterglow and host galaxy
contributions are subtracted and dust reddening is corrected for. Low-order
polynomial functions are fitted to the light curves. A K-correction is applied
to transform the light curves into the rest frame V band. GRB-SNe follow a
luminosity-decline rate relation similar to the Phillips relation for SNe Ia,
with $M_{V,peak} = 1.59^{+0.28}_{-0.24} \Delta m_{V,15} -
20.61^{+0.19}_{-0.22}$, with $\chi^2 = 5.2$ for 6 dof and $M_{V,peak}$ and
$\Delta m_{V,15}$ being the peak magnitude and decline rate in V band. This
luminosity-decline rate relation is tighter than the k-s relation, where k and
s are the factors describing the relative brightness and width to the light
curve of SN 1998bw. The peak luminosities of GRB-SNe are also correlated with
the time of peak: the brighter the GRB-SN, the longer the rise time. The light
curve of SN 1998bw stretched around the time of explosion can be used as a
template for GRB-SNe with reasonable confidence, but stretching around the peak
produces better results. The existence of such a relation provides a new
constraint on GRB explosion models. GRB-SNe can be used as standardizable
candles to measure cosmological distances and constrain cosmological
parameters.Comment: 17 pages, 15 figures. Submitted to Astronomy & Astrophysics on July
4, 201

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