5,760 research outputs found
General solution of the Poisson equation for Quasi-Birth-and-Death processes
We consider the Poisson equation , where
is the transition matrix of a Quasi-Birth-and-Death (QBD) process with
infinitely many levels, is a given infinite dimensional vector and is the unknown. Our main result is to provide the general solution of this
equation. To this purpose we use the block tridiagonal and block Toeplitz
structure of the matrix to obtain a set of matrix difference equations,
which are solved by constructing suitable resolvent triples
Hamiltonian analysis of subcritical stochastic epidemic dynamics
We extend a technique of approximation of the long-term behavior of a
supercritical stochastic epidemic model, using the WKB approximation and a
Hamiltonian phase space, to the subcritical case. The limiting behavior of the
model and approximation are qualitatively different in the subcritical case,
requiring a novel analysis of the limiting behavior of the Hamiltonian system
away from its deterministic subsystem. This yields a novel, general technique
of approximation of the quasistationary distribution of stochastic epidemic and
birth-death models, and may lead to techniques for analysis of these models
beyond the quasistationary distribution. For a classic SIS model, the
approximation found for the quasistationary distribution is very similar to
published approximations but not identical. For a birth-death process without
depletion of susceptibles, the approximation is exact. Dynamics on the phase
plane similar to those predicted by the Hamiltonian analysis are demonstrated
in cross-sectional data from trachoma treatment trials in Ethiopia, in which
declining prevalences are consistent with subcritical epidemic dynamics
Semigroup approach to birth-and-death stochastic dynamics in continuum
We describe a general approach to the construction of a state evolution
corresponding to the Markov generator of a spatial birth-and-death dynamics in
. We present conditions on the birth-and-death intensities which
are sufficient for the existence of an evolution as a strongly continuous
semigroup in a proper Banach space of correlation functions satisfying the
Ruelle bound. The convergence of a Vlasov-type scaling for the corresponding
stochastic dynamics is considered.Comment: 35 page
Markov evolutions and hierarchical equations in the continuum I. One-component systems
General birth-and-death as well as hopping stochastic dynamics of infinite
particle systems in the continuum are considered. We derive corresponding
evolution equations for correlation functions and generating functionals.
General considerations are illustrated in a number of concrete examples of
Markov evolutions appearing in applications.Comment: 47 page
Functional-integral based perturbation theory for the Malthus-Verhulst process
We apply a functional-integral formalism for Markovian birth and death
processes to determine asymptotic corrections to mean-field theory in the
Malthus-Verhulst process (MVP). Expanding about the stationary mean-field
solution, we identify an expansion parameter that is small in the limit of
large mean population, and derive a diagrammatic expansion in powers of this
parameter. The series is evaluated to fifth order using computational
enumeration of diagrams. Although the MVP has no stationary state, we obtain
good agreement with the associated {\it quasi-stationary} values for the
moments of the population size, provided the mean population size is not small.
We compare our results with those of van Kampen's -expansion, and apply
our method to the MVP with input, for which a stationary state does exist.Comment: 24 pages, 15 figure
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
Regulation mechanisms in spatial stochastic development models
The aim of this paper is to analyze different regulation mechanisms in
spatial continuous stochastic development models. We describe the density
behavior for models with global mortality and local establishment rates. We
prove that the local self-regulation via a competition mechanism (density
dependent mortality) may suppress a unbounded growth of the averaged density if
the competition kernel is superstable.Comment: 19 page
Uncoupled Analysis of Stochastic Reaction Networks in Fluctuating Environments
The dynamics of stochastic reaction networks within cells are inevitably
modulated by factors considered extrinsic to the network such as for instance
the fluctuations in ribsome copy numbers for a gene regulatory network. While
several recent studies demonstrate the importance of accounting for such
extrinsic components, the resulting models are typically hard to analyze. In
this work we develop a general mathematical framework that allows to uncouple
the network from its dynamic environment by incorporating only the
environment's effect onto the network into a new model. More technically, we
show how such fluctuating extrinsic components (e.g., chemical species) can be
marginalized in order to obtain this decoupled model. We derive its
corresponding process- and master equations and show how stochastic simulations
can be performed. Using several case studies, we demonstrate the significance
of the approach. For instance, we exemplarily formulate and solve a marginal
master equation describing the protein translation and degradation in a
fluctuating environment.Comment: 7 pages, 4 figures, Appendix attached as SI.pdf, under submissio
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