6,723 research outputs found
Exact Simulation of Non-stationary Reflected Brownian Motion
This paper develops the first method for the exact simulation of reflected
Brownian motion (RBM) with non-stationary drift and infinitesimal variance. The
running time of generating exact samples of non-stationary RBM at any time
is uniformly bounded by where is the
average drift of the process. The method can be used as a guide for planning
simulations of complex queueing systems with non-stationary arrival rates
and/or service time
Steady-state simulation of reflected Brownian motion and related stochastic networks
This paper develops the first class of algorithms that enable unbiased
estimation of steady-state expectations for multidimensional reflected Brownian
motion. In order to explain our ideas, we first consider the case of compound
Poisson (possibly Markov modulated) input. In this case, we analyze the
complexity of our procedure as the dimension of the network increases and show
that, under certain assumptions, the algorithm has polynomial-expected
termination time. Our methodology includes procedures that are of interest
beyond steady-state simulation and reflected processes. For instance, we use
wavelets to construct a piecewise linear function that can be guaranteed to be
within distance (deterministic) in the uniform norm to Brownian
motion in any compact time interval.Comment: Published at http://dx.doi.org/10.1214/14-AAP1072 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Extinction transitions in correlated external noise
We analyze the influence of long-range correlated (colored) external noise on
extinction phase transitions in growth and spreading processes. Uncorrelated
environmental noise (i.e., temporal disorder) was recently shown to give rise
to an unusual infinite-noise critical point [Europhys. Lett. 112, 30002
(2015)]. It is characterized by enormous density fluctuations that increase
without limit at criticality. As a result, a typical population decays much
faster than the ensemble average which is dominated by rare events. Using the
logistic evolution equation as an example, we show here that positively
correlated (red) environmental noise further enhances these effects. This
means, the correlations accelerate the decay of a typical population but slow
down the decay of the ensemble average. Moreover, the mean time to extinction
of a population in the active, surviving phase grows slower than a power law
with population size. To determine the complete critical behavior of the
extinction transition, we establish a relation to fractional random walks, and
we perform extensive Monte-Carlo simulations.Comment: 11 pages, 12 figures, Final versio
Fractional Brownian motion with a reflecting wall
Fractional Brownian motion, a stochastic process with long-time correlations
between its increments, is a prototypical model for anomalous diffusion. We
analyze fractional Brownian motion in the presence of a reflecting wall by
means of Monte Carlo simulations. While the mean-square displacement of the
particle shows the expected anomalous diffusion behavior , the interplay between the geometric confinement and the
long-time memory leads to a highly non-Gaussian probability density function
with a power-law singularity at the barrier. In the superdiffusive case,
, the particles accumulate at the barrier leading to a divergence of
the probability density. For subdiffusion, , in contrast, the
probability density is depleted close to the barrier. We discuss implications
of these findings, in particular for applications that are dominated by rare
events.Comment: 6 pages, 6 figures. Final version as publishe
From Equilibrium to Steady-State Dynamics after Switch-On of Shear
A relation between equilibrium, steady-state, and waiting-time dependent
dynamical two-time correlation functions in dense glass-forming liquids subject
to homogeneous steady shear flow is discussed. The systems under study show
pronounced shear thinning, i.e., a significant speedup in their steady-state
slow relaxation as compared to equilibrium. An approximate relation that
recovers the exact limit for small waiting times is derived following the
integration through transients (ITT) approach for the nonequilibrium
Smoluchowski dynamics, and is exemplified within a schematic model in the
framework of the mode-coupling theory of the glass transition (MCT). Computer
simulation results for the tagged-particle density correlation functions
corresponding to wave vectors in the shear-gradient directions from both
event-driven stochastic dynamics of a two-dimensional hard-disk system and from
previously published Newtonian-dynamics simulations of a three-dimensional
soft-sphere mixture are analyzed and compared with the predictions of the
ITT-based approximation. Good qualitative and semi-quantitative agreement is
found. Furthermore, for short waiting times, the theoretical description of the
waiting time dependence shows excellent quantitative agreement to the
simulations. This confirms the accuracy of the central approximation used
earlier to derive fluctuation dissipation ratios (Phys. Rev. Lett. 102,
135701). For intermediate waiting times, the correlation functions decay faster
at long times than the stationary ones. This behavior is predicted by our
theory and observed in simulations.Comment: 16 pages, 12 figures, submitted to Phys Rev
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