96 research outputs found
Time correlations for the parabolic Anderson model
We derive exact asymptotics of time correlation functions for the parabolic
Anderson model with homogeneous initial condition and time-independent tails
that decay more slowly than those of a double exponential distribution and have
a finite cumulant generating function. We use these results to give precise
asymptotics for statistical moments of positive order. Furthermore, we show
what the potential peaks that contribute to the intermittency picture look like
and how they are distributed in space. We also investigate for how long
intermittency peaks remain relevant in terms of ageing properties of the model.Comment: 28 page
Geometric characterization of intermittency in the parabolic Anderson model
We consider the parabolic Anderson problem on
with localized initial condition
and random i.i.d. potential . Under the assumption
that the distribution of has a double-exponential, or slightly
heavier, tail, we prove the following geometric characterization of
intermittency: with probability one, as , the overwhelming
contribution to the total mass comes from a slowly increasing
number of ``islands'' which are located far from each other. These ``islands''
are local regions of those high exceedances of the field in a box of side
length for which the (local) principal Dirichlet eigenvalue of the
random operator is close to the top of the spectrum in the box. We
also prove that the shape of in these regions is nonrandom and that
is close to the corresponding positive eigenfunction. This is the
geometric picture suggested by localization theory for the Anderson
Hamiltonian.Comment: Published at http://dx.doi.org/10.1214/009117906000000764 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment
We continue our study of the parabolic Anderson equation for the space-time field , where is the diffusion constant,
is the discrete Laplacian, is the coupling
constant, and is a space-time random
environment that drives the equation. The solution of this equation describes
the evolution of a "reactant" under the influence of a "catalyst" ,
both living on . In earlier work we considered three choices for :
independent simple random walks, the symmetric exclusion process, and the
symmetric voter model, all in equilibrium at a given density. We analyzed the
\emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the
successive moments of w.r.t.\ , and showed that these exponents
display an interesting dependence on the diffusion constant , with
qualitatively different behavior in different dimensions . In the present
paper we focus on the \emph{quenched} Lyapunov exponent, i.e., the exponential
growth rate of conditional on . We first prove existence and derive
some qualitative properties of the quenched Lyapunov exponent for a general
that is stationary and ergodic w.r.t.\ translations in and
satisfies certain noisiness conditions. After that we focus on the three
particular choices for mentioned above and derive some more detailed
properties. We close by formulating a number of open problems.Comment: In honour of J\"urgen G\"artner on the occasion of his 60th birthday,
33 pages. Final revised versio
A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems
In this paper, we propose a new method to compute the numerical flux of a finite volume scheme, used for the approximation of the solution of parabolic partial differential equation with nonlinear diffusion and convection terms a 1D, 2D or 3D domain. The nonlinear diffusion term be bounded away from zero except a finite number of values. The method is based on the solution, at each interface between two control volumes, of a nonlinear elliptic two point boundary value problem derived from the original equation with Dirichlet boundary conditions given by the values of the discrete approximation in both control volumes. We prove the existence of a solution to this two point boundary value problem. We show that the expression for the numerical flux can be yielded without referring to this solution. Furthermore, we prove that the so designed finite volume scheme has the expected stability properties and that its solution converges to the weak solution of the continuous problem. Numerical results show the increase of accuracy due to the use of this scheme, compared to some other schemes
Über das Teilentladungsverhalten von strombegrenzenden Hochspannungs-Hochleistungssicherungen
[no abstract
Stable limit laws for the parabolic Anderson model between quenched and annealed behaviour
Information-analytical systems as a basis of improving the efficiency of risk management
Building an effective system-risk management in an enterprise on the basis of integrated integration of risk management procedures into virtually all enterprise processes is associated with a wide range of tasks. Such integration processes can be simplified by using modern information technologies
Recommended from our members
Self-heating effects in organic semiconductor devices enhanced by positive temperature feedback
We studied the influence of heating effects in an organic device containing a layer sequence of n-doped / intrinsic / n-doped C60 between crossbar metal electrodes. A strong positive feedback between current and temperature occurs at high current densities beyond 100 A/cm2, as predicted by the extended Gaussian disorder model (EGDM) applicable to organic semiconductors. These devices give a perfect setting for studying the heat transport at high power densities because C60 can withstand temperatures above 200ʿ C. Infrared images of the device and detailed numerical simulations of the heat transport demonstrate that the electrical circuit produces a superposition of a homogeneous power dissipation in the active volume and strong heat sources localized at the contact edges ..
- …