The Reuter-Ledermann representation for birth and death processes

2012

We say that (weak/strong) Time Duality holds for continuous time quasi-birthand-death-processes if, starting from a fixed level, the first hitting time of the next upper level and the first hitting time of the next lower level have the same distribution. We present here a criterion for Time Duality in the case that transitions from one level to another have to pass through a given single state, so called bottleneck property. We also prove that a weaker form of reversibility called balanced under permutation is sufficient for Time Duality to hold. We then discuss the general case.

arXiv (Cornell University), 2017

We consider a class of birth-and-death processes describing a population made of d sub-populations of different types which interact with one another. The state space is Z d + (unbounded). We assume that the population goes almost surely to extinction, so that the unique stationary distribution is the Dirac measure at the origin. These processes are parametrized by a scaling parameter K which can be thought as the order of magnitude of the total size of the population at time 0. For any fixed finite time span, it is well-known that such processes, when renormalized by K, are close, in the limit K → +∞, to the solutions of a certain differential equation in R d + whose vector field is determined by the birth and death rates. We consider the case where there is a unique attractive fixed point (off the boundary of the positive orthant) for the vector field (while the origin is repulsive). What is expected is that, for K large, the process will stay in the vicinity of the fixed point for a very long time before being absorbed at the origin. To precisely describe this behavior, we prove the existence of a quasi-stationary distribution (qsd, for short). In fact, we establish a bound for the total variation distance between the process conditioned to non-extinction before time t and the qsd. This bound is exponentially small in t, for t log K. As a by-product, we obtain an estimate for the mean time to extinction in the qsd. We also quantify how close is the law of the process (not conditioned to non-extinction) either to the Dirac measure at the origin or to the qsd, for times much larger than log K and much smaller than the mean time to extinction, which is exponentially large as a function of K. Let us stress that we are interested in what happens for finite K. We obtain results much beyond what large deviation techniques could provide.

arXiv: Probability, 2015

In this paper we review some results on time-homogeneous birth-death processes. Specifically, for truncated birth-death processes with two absorbing or two reflecting endpoints, we recall the necessary and sufficient conditions on the transition rates such that the transition probabilities satisfy a spatial symmetry relation. The latter leads to simple expressions for first-passage-time densities and avoiding transition probabilities. This approach is thus thoroughly extended to the case of bilateral birth-death processes, even in the presence of catastrophes, and to the case of a two-dimensional birth-death process with constant rates.

2003

Journal de Physique, 1985

2014 On donne une formulation par intégrales de chemin du formalisme de Fock pour objets classiques, premièrement introduit par Doi, et on l'applique à des processus généraux de naissance et mort sur réseau. L'introduction de variables auxiliaires permet de donner une forme Markovienne aux lois d'évolution des chemins aléatoires avec mémoire et des processus irréversibles d'agrégation. Les théories des champs existantes pour ces processus sont obtenues dans la limite continue. On discute brièvement des implications de cette méthode pour leur comportement asymptotique.

Queueing Systems, 2006

In this paper we consider nonhomogeneous birth and death processes (BDP) with periodic rates. Two important parameters are studied, which are helpful to describe a nonhomogeneous BDP X = X (t), t ≥ 0: the limiting mean value (namely, the mean length of the queue at a given time t) and the double mean (i.e. the mean length of the queue for the whole duration of the BDP). We find conditions of existence of the means and determine bounds for their values, involving also the truncated BDP X N . Finally we present some examples where these bounds are used in order to approximate the double mean.

Journal of Statistical Physics, 1976

We derive the path integral representation of the conditional probability for a Markovian process starting from the master equation. Existing derivations require both the variable and the transition probability to be extensive. We show that this requirement may be relaxed if Langer's formula for the transition probability is used. We prove that different path integral representations appearing in the literature are in fact equivalent and correspond to various choices of an arbitrary parameter.

Problems of Information Transmission, 2018

2008

We consider ordinary and conditional first passage times in a general birth-death process. Under existence conditions, we derive closed-form expressions for the kth order moment of the defined random variables, k ≥ 1. We also give an explicit condition for a birth-death process to be ergodic degree 3. Based on the obtained results, we analyze some applications for Markovian queueing systems. In particular, we compute for some non-standard Markovian queues, the moments of the busy period duration, the busy cycle duration, and the state-dependent waiting time in queue.

Mathematical Modelling of Natural Phenomena

Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of infinite, coupled ordinary differential equations (the so-called master equation) describes the time-dependence of the probability of each system state. Using a generating function, the master equation can be transformed into a partial differential equation. In this contribution we analyze the integrability of two types of stochastic birth-death processes (with polynomial birth and death rates) using standard differential Galois theory. We discuss the integrability of the PDE via a Laplace transform acting over the temporal variable. We show that the PDE is not integrable except for the case in which rates are linear functions of the number of individuals.