Ergodicity and regularity of invariant measure for branching Markov processes with immigration

In this thesis we consider systems of finitely many particles moving on paths given by a strong Markov process and undergoing branching and reproduction at random times. The branching rate of a particle, its number of offspring and their spatial distribution are allowed to depend on the particle's position and possibly on the configuration of coexisting particles. In addition there is immigration of new particles, with the rate of immigration and the distribution of immigrants possibly depending on the configuration of pre-existing particles as well. In the first two chapters of this work, we concentrate on the case that the joint motion of particles is governed by a diffusion with interacting components. The resulting process of particle configurations was studied by E. Löcherbach (2002, 2004) and is known as a branching diffusion with immigration (BDI). Chapter 1 contains a detailed introduction of the basic model assumptions, in particular an assumption of ergodicity which guarantees that the BDI process is positive Harris recurrent with finite invariant measure on the configuration space. This object and a closely related quantity, namely the invariant occupation measure on the single-particle space, are investigated in Chapter 2 where we study the problem of the existence of Lebesgue-densities with nice regularity properties. For example, it turns out that the existence of a continuous density for the invariant measure depends on the mechanism by which newborn particles are distributed in space, namely whether branching particles reproduce at their death position or their offspring are distributed according to an absolutely continuous transition kernel. In Chapter 3, we assume that the quantities defining the model depend only on the spatial position but not on the configuration of coexisting particles. In this framework (which was considered by Höpfner and Löcherbach (2005) in the special case that branching particles reproduce at their death position), the particle motions are independent, and we can allow for more general Markov processes instead of diffusions. The resulting configuration process is a branching Markov process in the sense introduced by Ikeda, Nagasawa and Watanabe (1968), complemented by an immigration mechanism. Generalizing results obtained by Höpfner and Löcherbach (2005), we give sufficient conditions for ergodicity in the sense of positive recurrence of the configuration process and finiteness of the invariant occupation measure in the case of general particle motions and offspring distributions.

Modeling the survival of T-cell lymphocytes using compound branching processes

Radiotherapy has been a method of choice in cancer treatment for a number of years. Mathematical modeling is an important tool in studying the survival behavior of any cell as well as its radiosensitivity. One particular cell under investigation is the normal T-cell, the radiosensitivity of which may be indicative to the patient's tolerance to radiation doses.^ The model derived is a compound branching process with a random initial population of T-cells that is assumed to have compound distribution. T-cells in any generation are assumed to double or die at random lengths of time. This population is assumed to undergo a random number of generations within a period of time. The model is then used to obtain an estimate for the survival probability of T-cells for the data under investigation. This estimate is derived iteratively by applying the likelihood principle. Further assessment of the validity of the model is performed by simulating a number of subjects under this model.^ This study shows that there is a great deal of variation in T-cells survival from one individual to another. These variations can be observed under normal conditions as well as under radiotherapy. The findings are in agreement with a recent study and show that genetic diversity plays a role in determining the survival of T-cells. ^

Branching angles of pyramidal cell dendrites follow common geometrical design principles in different cortical areas

Unraveling pyramidal cell structure is crucial to understanding cortical circuit computations. Although it is well known that pyramidal cell branching structure differs in the various cortical areas, the principles that determine the geometric shapes of these cells are not fully understood. Here we analyzed and modeled with a von Mises distribution the branching angles in 3D reconstructed basal dendritic arbors of hundreds of intracellularly injected cortical pyramidal cells in seven different cortical regions of the frontal, parietal, and occipital cortex of the mouse. We found that, despite the differences in the structure of the pyramidal cells in these distinct functional and cytoarchitectonic cortical areas, there are common design principles that govern the geometry of dendritic branching angles of pyramidal cells in all cortical areas.

Dual pathways for regulation of root branching by nitrate

Root development is extremely sensitive to variations in nutrient supply, but the mechanisms are poorly understood. We have investigated the processes by which nitrate (NO3−), depending on its availability and distribution, can have both positive and negative effects on the development and growth of lateral roots. When Arabidopsis roots were exposed to a locally concentrated supply of NO3− there was no increase in lateral root numbers within the NO3−-rich zone, but there was a localized 2-fold increase in the mean rate of lateral root elongation, which was attributable to a corresponding increase in the rate of cell production in the lateral root meristem. Localized applications of other N sources did not stimulate lateral root elongation, consistent with previous evidence that the NO3− ion is acting as a signal rather than a nutrient. The axr4 auxin-resistant mutant was insensitive to the stimulatory effect of NO3−, suggesting an overlap between the NO3− and auxin response pathways. High rates of NO3− supply to the roots had a systemic inhibitory effect on lateral root development that acted specifically at the stage when the laterals had just emerged from the primary root, apparently delaying final activation of the lateral root meristem. A nitrate reductase-deficient mutant showed increased sensitivity to this systemic inhibitory effect, suggesting that tissue NO3− levels may play a role in generating the inhibitory signal. We present a model in which root branching is modulated by opposing signals from the plant’s internal N status and the external supply of NO3−.

Flt1 acts as a negative regulator of tip cell formation and branching morphogenesis in the zebrafish embryo

Endothelial tip cells guide angiogenic sprouts by exploring the local environment for guidance cues such as vascular endothelial growth factor (VegfA). Here we present Flt1 (Vegf receptor 1) loss- and gain-of-function data in zebrafish showing that Flt1 regulates tip cell formation and arterial branching morphogenesis. Zebrafish embryos expressed soluble Flt1 (sFlt1) and membrane-bound Flt1 (mFlt1). In Tg(flt1(BAC):yfp) × Tg(kdrl:ras-cherry)(s916) embryos, flt1:yfp was expressed in tip, stalk and base cells of segmental artery sprouts and overlapped with kdrl:cherry expression in these domains. flt1 morphants showed increased tip cell numbers, enhanced angiogenic behavior and hyperbranching of segmental artery sprouts. The additional arterial branches developed into functional vessels carrying blood flow. In support of a functional role for the extracellular VEGF-binding domain of Flt1, overexpression of sflt1 or mflt1 rescued aberrant branching in flt1 morphants, and overexpression of sflt1 or mflt1 in controls resulted in short arterial sprouts with reduced numbers of filopodia. flt1 morphants showed reduced expression of Notch receptors and of the Notch downstream target efnb2a, and ectopic expression of flt4 in arteries, consistent with loss of Notch signaling. Conditional overexpression of the notch1a intracellular cleaved domain in flt1 morphants restored segmental artery patterning. The developing nervous system of the trunk contributed to the distribution of Flt1, and the loss of flt1 affected neurons. Thus, Flt1 acts in a Notch-dependent manner as a negative regulator of tip cell differentiation and branching. Flt1 distribution may be fine-tuned, involving interactions with the developing nervous system.

On the Maximum of a Branching Process Conditioned on the Total Progeny

The maximum M of a critical Bienaymé-Galton-Watson process conditioned on the total progeny N is studied. Imbedding of the process in a random walk is used. A limit theorem for the distribution of M as N → ∞ is proved. The result is trasferred to the non-critical processes. A corollary for the maximal strata of a random rooted labeled tree is obtained.

On the Structure of Spatial Branching Processes

The paper is a contribution to the theory of branching processes with discrete time and a general phase space in the sense of [2]. We characterize the class of regular, i.e. in a sense sufficiently random, branching processes (Φk) k∈Z by almost sure properties of their realizations without making any assumptions about stationarity or existence of moments. This enables us to classify the clans of (Φk) into the regular part and the completely non-regular part. It turns out that the completely non-regular branching processes are built up from single-line processes, whereas the regular ones are mixtures of left-tail trivial processes with a Poisson family structure.

Branching Processes with Immigration and Integer-valued Time Series

In this paper, we indicate how integer-valued autoregressive time series Ginar(d) of ordre d, d ≥ 1, are simple functionals of multitype branching processes with immigration. This allows the derivation of a simple criteria for the existence of a stationary distribution of the time series, thus proving and extending some results by Al-Osh and Alzaid [1], Du and Li [9] and Gauthier and Latour [11]. One can then transfer results on estimation in subcritical multitype branching processes to stationary Ginar(d) and get consistency and asymptotic normality for the corresponding estimators. The technique covers autoregressive moving average time series as well.

Robust Parametric Estimation of Branching Processes with a Random Number of Ancestors

2000 Mathematics Subject Classification: 60J80.

A Limit Theorem for Multi-Type Subcritical Age-Dependent Branching Processes with two Types of Immigration

Марусия Н. Славчова-Божкова - В настоящата работа се обобщава една гранична теорема за докритичен многомерен разклоняващ се процес, зависещ от възрастта на частиците с два типа имиграция. Целта е да се обобщи аналогичен резултат в едномерния случай като се прилагат “coupling” метода, теория на възстановяването и регенериращи процеси.

Bootstrap for Critical Branching Process with Non-Stationary Immigration

2000 Mathematics Subject Classification: Primary 60J80, Secondary 62F12, 60G99.

Option Pricing by Branching Process

2000 Mathematics Subject Classification: 60J80, 62P05.

Branching Populations of Cells Bearing a Continuous Label

2000 Mathematics Subject Classification: 60J80.

Comparison between Numerical and Simulation Methods for Age-dependent Branching Models with Immigration

2000 Mathematics Subject Classification: primary: 60J80, 60J85, secondary: 62M09, 92D40

Parametric Estimation in Branching Processes with an Increasing Random Number of Ancestors

2000 Mathematics Subject Classification: 60J80, 62M05