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A Note on the Asymptotic Behaviour of a Periodic Multitype Galton-Watson Branching Process

González, M., Martínez, R., Mota, M. (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 60J80.In this work, the problem of the limiting behaviour of an irreducible Multitype Galton-Watson Branching Process with period d greater than 1 is considered. More specifically, almost sure convergence of some linear functionals depending on d consecutive generations is studied under hypothesis of non extinction. As consequence the main parameters of the model are given a convenient interpretation from a practical point of view. For a better understanding...

Adaptive dynamics in logistic branching populations

Nicolas Champagnat, Amaury Lambert (2008)

Banach Center Publications

The biological theory of adaptive dynamics proposes a description of the long-time evolution of an asexual population, based on the assumptions of large population, rare mutations and small mutation steps. Under these assumptions, the evolution of a quantitative dominant trait in an isolated population is described by a deterministic differential equation called 'canonical equation of adaptive dynamics'. In this work, in order to include the effect of genetic drift in this model, we consider instead...

Ancestral processes with selection: Branching and Moran models

Ellen Baake, Robert Bialowons (2008)

Banach Center Publications

We consider two versions of stochastic population models with mutation and selection. The first approach relies on a multitype branching process; here, individuals reproduce and change type (i.e., mutate) independently of each other, without restriction on population size. We analyse the equilibrium behaviour of this model, both in the forward and in the backward direction of time; the backward point of view emerges if the ancestry of individuals chosen randomly from the present population is traced...

Approximate Aggregation Methods in Discrete Time Stochastic Population Models

L. Sanz, J. A. Alonso (2010)

Mathematical Modelling of Natural Phenomena

Approximate aggregation techniques consist of introducing certain approximations that allow one to reduce a complex system involving many coupled variables obtaining a simpler ʽʽaggregated systemʼʼ governed by a few variables. Moreover, they give results that allow one to extract information about the complex original system in terms of the behavior of the reduced one. Often, the feature that allows one to carry out such a reduction is the presence...

Branching brownian motion with an inhomogeneous breeding potential

J. W. Harris, S. C. Harris (2009)

Annales de l'I.H.P. Probabilités et statistiques

This article concerns branching brownian motion (BBM) with dyadic branching at rate β|y|p for a particle with spatial position y∈ℝ, where β>0. It is known that for p>2 the number of particles blows up almost surely in finite time, while for p=2 the expected number of particles alive blows up in finite time, although the number of particles alive remains finite almost surely, for all time. We define the right-most particle, Rt, to be the supremum of the spatial positions of the particles...

Branching random motions, nonlinear hyperbolic systems and travellind waves

Nikita Ratanov (2006)

ESAIM: Probability and Statistics

A branching random motion on a line, with abrupt changes of direction, is studied. The branching mechanism, being independent of random motion, and intensities of reverses are defined by a particle's current direction. A solution of a certain hyperbolic system of coupled non-linear equations (Kolmogorov type backward equation) has a so-called McKean representation via such processes. Commonly this system possesses travelling-wave solutions. The convergence of solutions with Heaviside terminal...

Changing the branching mechanism of a continuous state branching process using immigration

Romain Abraham, Jean-François Delmas (2009)

Annales de l'I.H.P. Probabilités et statistiques

We consider an initial population whose size evolves according to a continuous state branching process. Then we add to this process an immigration (with the same branching mechanism as the initial population), in such a way that the immigration rate is proportional to the whole population size. We prove this continuous state branching process with immigration proportional to its own size is itself a continuous state branching process. By considering the immigration as the apparition of a new type,...

Coupling a branching process to an infinite dimensional epidemic process***

Andrew D. Barbour (2010)

ESAIM: Probability and Statistics

Branching process approximation to the initial stages of an epidemic process has been used since the 1950's as a technique for providing stochastic counterparts to deterministic epidemic threshold theorems. One way of describing the approximation is to construct both branching and epidemic processes on the same probability space, in such a way that their paths coincide for as long as possible. In this paper, it is shown, in the context of a Markovian model of parasitic infection, that coincidence...

Lévy processes conditioned on having a large height process

Mathieu Richard (2013)

Annales de l'I.H.P. Probabilités et statistiques

In the present work, we consider spectrally positive Lévy processes ( X t , t 0 ) not drifting to + and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with X ) before hitting 0 . This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law x of this conditioned process (starting at x g t ; 0 ) is defined as a Doob h -transform via a martingale. For Lévy processes with infinite variation paths, this martingale...

Nonlinear filtering for observations on a random vector field along a random path. Application to atmospheric turbulent velocities

Christophe Baehr (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

To filter perturbed local measurements on a random medium, a dynamic model jointly with an observation transfer equation are needed. Some media given by PDE could have a local probabilistic representation by a Lagrangian stochastic process with mean-field interactions. In this case, we define the acquisition process of locally homogeneous medium along a random path by a Lagrangian Markov process conditioned to be in a domain following the path and conditioned to the observations. The nonlinear...

Perturbing transient random walk in a random environment with cookies of maximal strength

Elisabeth Bauernschubert (2013)

Annales de l'I.H.P. Probabilités et statistiques

We consider a left-transient random walk in a random environment on that will be disturbed by cookies inducing a drift to the right of strength 1. The number of cookies per site is i.i.d. and independent of the environment. Criteria for recurrence and transience of the random walk are obtained. For this purpose we use subcritical branching processes in random environments with immigration and formulate criteria for recurrence and transience for these processes.

Small-time behavior of beta coalescents

Julien Berestycki, Nathanaël Berestycki, Jason Schweinsberg (2008)

Annales de l'I.H.P. Probabilités et statistiques

For a finite measure Λ on [0, 1], the Λ-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate ∫01xk−2(1−x)b−kΛ(dx). It has recently been shown that if 1<α<2, the Λ-coalescent in which Λ is the Beta (2−α, α) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an α-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time...

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