40,331 research outputs found
A martingale analysis of first passage times of time-dependent Wiener diffusion models
Research in psychology and neuroscience has successfully modeled decision
making as a process of noisy evidence accumulation to a decision bound. While
there are several variants and implementations of this idea, the majority of
these models make use of a noisy accumulation between two absorbing boundaries.
A common assumption of these models is that decision parameters, e.g., the rate
of accumulation (drift rate), remain fixed over the course of a decision,
allowing the derivation of analytic formulas for the probabilities of hitting
the upper or lower decision threshold, and the mean decision time. There is
reason to believe, however, that many types of behavior would be better
described by a model in which the parameters were allowed to vary over the
course of the decision process.
In this paper, we use martingale theory to derive formulas for the mean
decision time, hitting probabilities, and first passage time (FPT) densities of
a Wiener process with time-varying drift between two time-varying absorbing
boundaries. This model was first studied by Ratcliff (1980) in the two-stage
form, and here we consider the same model for an arbitrary number of stages
(i.e. intervals of time during which parameters are constant). Our calculations
enable direct computation of mean decision times and hitting probabilities for
the associated multistage process. We also provide a review of how martingale
theory may be used to analyze similar models employing Wiener processes by
re-deriving some classical results. In concert with a variety of numerical
tools already available, the current derivations should encourage mathematical
analysis of more complex models of decision making with time-varying evidence
A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes
We consider products of random matrices that are small, independent
identically distributed perturbations of a fixed matrix . Focusing on the
eigenvalues of of a particular size we obtain a limit to a SDE in a
critical scaling. Previous results required to be a (conjugated) unitary
matrix so it could not have eigenvalues of different modulus. From the result
we can also obtain a limit SDE for the Markov process given by the action of
the random products on the flag manifold. Applying the result to random
Schr\"odinger operators we can improve some result by Valko and Virag showing
GOE statistics for the rescaled eigenvalue process of a sequence of Anderson
models on long boxes. In particular we solve a problem posed in their work.Comment: new version, parts rearrange
Poisson's equation for discrete-time quasi-birth-and-death processes
We consider Poisson's equation for quasi-birth-and-death processes (QBDs) and
we exploit the special transition structure of QBDs to obtain its solutions in
two different forms. One is based on a decomposition through first passage
times to lower levels, the other is based on a recursive expression for the
deviation matrix.
We revisit the link between a solution of Poisson's equation and perturbation
analysis and we show that it applies to QBDs. We conclude with the PH/M/1 queue
as an illustrative example, and we measure the sensitivity of the expected
queue size to the initial value
Large-time behavior in non-symmetric Fokker-Planck equations
We consider three classes of linear non-symmetric Fokker-Planck equations
having a unique steady state and establish exponential convergence of solutions
towards the steady state with explicit (estimates of) decay rates. First,
"hypocoercive" Fokker-Planck equations are degenerate parabolic equations such
that the entropy method to study large-time behavior of solutions has to be
modified. We review a recent modified entropy method (for non-symmetric
Fokker-Planck equations with drift terms that are linear in the position
variable). Second, kinetic Fokker-Planck equations with non-quadratic
potentials are another example of non-symmetric Fokker-Planck equations. Their
drift term is nonlinear in the position variable. In case of potentials with
bounded second-order derivatives, the modified entropy method allows to prove
exponential convergence of solutions to the steady state. In this application
of the modified entropy method symmetric positive definite matrices solving a
matrix inequality are needed. We determine all such matrices achieving the
optimal decay rate in the modified entropy method. In this way we prove the
optimality of previous results. Third, we discuss the spectral properties of
Fokker-Planck operators perturbed with convolution operators. For the
corresponding Fokker-Planck equation we show existence and uniqueness of a
stationary solution. Then, exponential convergence of all solutions towards the
stationary solution is proven with an uniform rate
Space-charge distortion of transverse profiles measured by electron-based Ionization Profile Monitors and correction methods
Measurements of transverse profiles using Ionization Profile Monitors (IPMs)
for high brightness beams are affected by the electromagnetic field of the
beam. This interaction may cause a distortion of the measured profile shape
despite strong external magnetic field applied to impose limits on the
transverse movement of electrons. The mechanisms leading to this distortion are
discussed in detail. The distortion itself is described by means of analytic
calculations for simplified beam distributions and a full simulation model for
realistic distributions. Simple relation for minimum magnetic field scaling
with beam parameters for avoiding profile distortions is presented. Further,
application of machine learning algorithms to the problem of reconstructing the
actual beam profile from distorted measured profile is presented. The obtained
results show good agreement for tests on simulation data. The performance of
these algorithms indicate that they could be very useful for operations of IPMs
on high brightness beams or IPMs with weak magnetic field
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