6,390 research outputs found
Measuring the Initial Transient: Reflected Brownian Motion
We analyze the convergence to equilibrium of one-dimensional reflected
Brownian motion (RBM) and compute a number of related initial transient
formulae. These formulae are of interest as approximations to the initial
transient for queueing systems in heavy traffic, and help us to identify
settings in which initialization bias is significant. We conclude with a
discussion of mean square error for RBM. Our analysis supports the view that
initial transient effects for RBM and related models are typically of modest
size relative to the intrinsic stochastic variability, unless one chooses an
especially poor initialization.Comment: 14 pages, 3 figure
Quantitative estimates for the long time behavior of an ergodic variant of the telegraph process
Motivated by stability questions on piecewise deterministic Markov models of
bacterial chemotaxis, we study the long time behavior of a variant of the
classic telegraph process having a non-constant jump rate that induces a drift
towards the origin. We compute its invariant law and show exponential
ergodicity, obtaining a quantitative control of the total variation distance to
equilibrium at each instant of time. These results rely on an exact description
of the excursions of the process away from the origin and on the explicit
construction of an original coalescent coupling for both velocity and position.
Sharpness of the obtained convergence rate is discussed.Comment: Definitive version of former paper "Quantitative estimates for the
long time behavior of a PDMP describing the movement of bacteria", now
accepted in Advances in Applied Probability. Presentation changed. A
diffusive scaling limit result is added. Sharpness of the long-time
convergence rate is discussed. 20 pages, 3 figure
Super-hydrodynamic limit in interacting particle systems
This paper is a follow-up of the work initiated in [3], where it has been
investigated the hydrodynamic limit of symmetric independent random walkers
with birth at the origin and death at the rightmost occupied site. Here we
obtain two further results: first we characterize the stationary states on the
hydrodynamic time scale and show that they are given by a family of linear
macroscopic profiles whose parameters are determined by the current reservoirs
and the system mass. Then we prove the existence of a super-hyrdrodynamic time
scale, beyond the hydrodynamic one. On this larger time scale the system mass
fluctuates and correspondingly the macroscopic profile of the system randomly
moves within the family of linear profiles, with the randomness of a Brownian
motion.Comment: 22 page
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