5,595 research outputs found
Taylor expansions and Castell estimates for solutions of stochastic differential equations driven by rough paths
We study the Taylor expansion for the solutions of differential equations
driven by -rough paths with . We prove a general theorem concerning the
convergence of the Taylor expansion on a nonempty interval provided that the
vector fields are analytic on a ball centered at the initial point. We also
derive criteria that enable us to study the rate of convergence of the Taylor
expansion. Finally and this is also the main and the most original part of this
paper, we prove Castell expansions and tail estimates with exponential decays
for the remainder terms of the solutions of the stochastic differential
equations driven by continuous centered Gaussian process with finite
variation and fractional Brownian motion with Hurst parameter
.Comment: Final version for publis
Inhomogeneous systematic signals in cosmic shear observations
We calculate the systematic errors in the weak gravitational lensing power
spectrum which would be caused by spatially varying calibration (i.e.
multiplicative) errors, such as might arise from uncorrected seeing or
extinction variations. The systematic error is fully described by the angular
two-point correlation function of the systematic in the case of the 2D lensing
that we consider here. We investigate three specific cases: Gaussian,
``patchy'' and exponential correlation functions. In order to keep systematic
errors below statistical errors in future LSST-like surveys, the spatial
variation of calibration should not exceed 3% rms. This conclusion is
independently true for all forms of correlation function we consider. The
relative size the E- and B-mode power spectrum errors does, however, depend
upon the form of the correlation function, indicating that one cannot repair
the E-mode power spectrum systematics by means of the B-mode measurements.Comment: 8 pages, 3 figures. Changes reflect PRD published versio
Speckle from phase ordering systems
The statistical properties of coherent radiation scattered from
phase-ordering materials are studied in detail using large-scale computer
simulations and analytic arguments. Specifically, we consider a two-dimensional
model with a nonconserved, scalar order parameter (Model A), quenched through
an order-disorder transition into the two-phase regime. For such systems it is
well established that the standard scaling hypothesis applies, consequently the
average scattering intensity at wavevector _k and time t' is proportional to a
scaling function which depends only on a rescaled time, t ~ |_k|^2 t'. We find
that the simulated intensities are exponentially distributed, with the
time-dependent average well approximated using a scaling function due to Ohta,
Jasnow, and Kawasaki. Considering fluctuations around the average behavior, we
find that the covariance of the scattering intensity for a single wavevector at
two different times is proportional to a scaling function with natural
variables mt = |t_1 - t_2| and pt = (t_1 + t_2)/2. In the asymptotic large-pt
limit this scaling function depends only on z = mt / pt^(1/2). For small values
of z, the scaling function is quadratic, corresponding to highly persistent
behavior of the intensity fluctuations. We empirically establish a connection
between the intensity covariance and the two-time, two-point correlation
function of the order parameter. This connection allows sensitive testing,
either experimental or numerical, of existing theories for two-time
correlations in systems undergoing order-disorder phase transitions. Comparison
between theory and our numerical results requires no adjustable parameters.Comment: 18 pgs RevTeX, to appear in PR
A simple method for the existence of a density for stochastic evolutions with rough coefficients
We extend the validity of a simple method for the existence of a density for
stochastic differential equations, first introduced in [DebRom2014], by proving
local estimate for the density, existence for the density with summable drift,
and by improving the regularity of the density.Comment: Added a section with examples and application
Extended Kramers-Moyal analysis applied to optical trapping
The Kramers-Moyal analysis is a well established approach to analyze
stochastic time series from complex systems. If the sampling interval of a
measured time series is too low, systematic errors occur in the analysis
results. These errors are labeled as finite time effects in the literature. In
the present article, we present some new insights about these effects and
discuss the limitations of a previously published method to estimate
Kramers-Moyal coefficients at the presence of finite time effects. To increase
the reliability of this method and to avoid misinterpretations, we extend it by
the computation of error estimates for estimated parameters using a Monte Carlo
error propagation technique. Finally, the extended method is applied to a data
set of an optical trapping experiment yielding estimations of the forces acting
on a Brownian particle trapped by optical tweezers. We find an increased
Markov-Einstein time scale of the order of the relaxation time of the process
which can be traced back to memory effects caused by the interaction of the
particle and the fluid. Above the Markov-Einstein time scale, the process can
be very well described by the classical overdamped Markov model for Brownian
motion.Comment: 14 pages, 18 figure
- …