4,909 research outputs found
Intertwining of birth-and-death processes
It has been known for a long time that for birth-and-death processes started
in zero the first passage time of a given level is distributed as a sum of
independent exponentially distributed random variables, the parameters of which
are the negatives of the eigenvalues of the stopped process. Recently, Diaconis
and Miclo have given a probabilistic proof of this fact by constructing a
coupling between a general birth-and-death process and a process whose birth
rates are the negatives of the eigenvalues, ordered from high to low, and whose
death rates are zero, in such a way that the latter process is always ahead of
the former, and both arrive at the same time at the given level. In this note,
we extend their methods by constructing a third process, whose birth rates are
the negatives of the eigenvalues ordered from low to high and whose death rates
are zero, which always lags behind the original process and also arrives at the
same time.Comment: 12 pages. 1 figure. Some typoes corrected and minor change
Subcritical contact processes seen from a typical infected site
What is the long-time behavior of the law of a contact process started with a
single infected site, distributed according to counting measure on the lattice?
This question is related to the configuration as seen from a typical infected
site and gives rise to the definition of so-called eigenmeasures, which are
possibly infinite measures on the set of nonempty configurations that are
preserved under the dynamics up to a multiplicative constant. In this paper, we
study eigenmeasures of contact processes on general countable groups in the
subcritical regime. We prove that in this regime, the process has a unique
spatially homogeneous eigenmeasure. As an application, we show that the
exponential growth rate is continuously differentiable and strictly decreasing
as a function of the recovery rate, and we give a formula for the derivative in
terms of the eigenmeasures of the contact process and its dual.Comment: Changed the organization of the proofs somewhat to more clearly make
a link to classical results about quasi-invariant laws. 44 page
Trimmed trees and embedded particle systems
In a supercritical branching particle system, the trimmed tree consists of
those particles which have descendants at all times. We develop this concept in
the superprocess setting. For a class of continuous superprocesses with Feller
underlying motion on compact spaces, we identify the trimmed tree, which turns
out to be a binary splitting particle system with a new underlying motion that
is a compensated h-transform of the old one. We show how trimmed trees may be
estimated from above by embedded binary branching particle systems.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000009
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