Motivated as a null model for comparison with data, we study the following model for a phylogenetic tree on $n$ extant species. The origin of the clade is a random time in the past, whose (improper) distribution is uniform on $(0,\infty)$. After that origin, the process of extinctions and speciations is a continuous-time critical branching process of constant rate, conditioned on having the prescribed number $n$ of species at the present time. We study various mathematical properties of this model as $n \to \infty$ limits: time of origin and of most recent common ancestor; pattern of divergence times within lineage trees; time series of numbers of species; number of extinct species in total, or ancestral to extant species; and "local" structure of the tree itself. We emphasize several mathematical techniques: associating walks with trees, a point process representation of lineage trees, and Brownian limits.
We continue our study of critical branching random walk and branching capacity. In this paper we introduce branching recurrence and branching transience and prove an analogous version of Wiener's Test.
We extend the theory of discrete capacity to critical branching random walk. We introduce branching capacity for any finite subset of $\Z^d, d\geq5$. Analogous to the regular discrete capacity, branching capacity is closely related to the asymptotics of the probability of visiting a fixed finite set by a critical branching random walk starting from a distant point and the conditional distribution of the hitting point.
We derive some additional results on the Bienyamé-Galton-Watson branching process with $θ-$linear fractional branching mechanism, as studied in \cite{Sag}. This includes: the explicit expression of the limit laws in both the sub-critical cases and the super-critical cases with finite mean, the long-run behavior of the population size in the critical case, limit laws in the super-critical cases with infinite mean when the $θ$-process is either regular or explosive, results regarding the time to absorption, an expression of the probability law of the $θ$-branching mechanism involving Bell polyno…
We study the branching random walk on weighted graphs; site-breeding and edge-breeding branching random walks on graphs are seen as particular cases. We describe the strong critical value in terms of a geometrical parameter of the graph. We characterize the weak critical value and relate it to another geometrical parameter. We prove that, at the strong critical value, the process dies out locally almost surely; while, at the weak critical value, global survival and global extinction are both possible.
In this paper, we study the critical branching random walk in the critical dimension, $Z^4$. We provide the asymptotics of the probability of visiting a fixed finite subset and the range of the critical branching random walk conditioned on the total number of offsprings. We also prove that conditioned on visiting, the first visiting point converges in distribution.
We define the height process for super-critical continuous state branching processes with quadratic branching mechanism. It appears as a projective limit of Brownian motions with positive drift reflected at 0 and a>0 as a goes to infinity. Then we extend the pruning procedure of branching processes to the super-critical case. This give a complete duality picture between pruning and size proportional immigration for quadratic continuous state branching processes.
This survey aims at collecting and presenting results for one-type, discrete time branching processes with random control functions. In particular, the subclass of critical migration processes with different regimes of immigration and emigration is reviewed in detail. Critical controlled branching processes with continuous state space are also discussed.
This paper studies: (i) the long time behaviour of the empirical distribution of age and normalised position of an age dependent critical branching Markov process conditioned on non-extinction; and (ii) the super-process limit of a sequence of age dependent critical branching Brownian motions.
We consider a branching stable process with positive jumps, i.e. a continuous-time branching process in which the particles evolve independently as stable L{é}vy processes with positive jumps. Assuming the branching mechanism is critical or subcritical, we compute the asymptotics of the maximum location ever reached by a particle of the process.
We observe the Galton-Watson Branching Processes. Limit properties of transition functions and their convergence to invariant measures are investigated.
Given a branching random walk on a graph, we consider two kinds of truncations: by inhibiting the reproduction outside a subset of vertices and by allowing at most $m$ particles per site. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.
We describe some connections, via composition, between two functional spaces: the space of (sub)critical branching mechanisms and the space of Bernstein functions. The functions ${\bf e}_α: x\to x^α$ where $x\geq0$ and $0<α\leq 1/2$, and in particular the critical parameter $α=1/2$, play a distinguished role.
Critical branching processes in a varying environment behave much the same as critical Galton-Watson processes. In this note we like to confirm this finding with regard to the underlying genealogical structures. In particular, we consider the most recent common ancestor given survival and the corresponding reduced branching processes, in the spirit of Zubkov (1975) and Fleischmann and Siegmund-Schultze (1977).
In this paper, we study the probability of visiting a distant point $a\in \mathbb{Z}^4$ by critical branching random walk starting from the origin. We prove that this probability is bounded by $1/(|a|^2\log |a|)$ up to a constant.
Let (Z n) n$\ge$0 with Z n = (Z n (i, j)) 1$\le$i,j$\le$p be a p multi-type critical branching process in random environment, and let M n be the expectation of Z n given a fixed environment. We prove theorems on convergence in distribution of sequences of branching processes Zn |Mn| /|Z n | > 0 and ln Zn $\sqrt$ n /|Z n | > 0. These theorems extend similar results for single-type critical branching process in random environment.
Under natural assumptions a Feller type diffusion approximation is derived for critical multi-type branching processes with immigration when the offspring mean matrix is primitive (in other words, positively regular). Namely, it is proved that a sequence of appropriately scaled random step functions formed from a sequence of critical primitive multi-type branching processes with immigration converges weakly towards a squared Bessel process supported by a ray determined by the Perron vector of the offspring mean matrix.
Under natural assumptions, a Feller type diffusion approximation is derived for critical, irreducible multi-type continuous state and continuous time branching processes with immigration. Namely, it is proved that a sequence of appropriately scaled random step functions formed from a critical, irreducible multi-type continuous state and continuous time branching process with immigration converges weakly towards a squared Bessel process supported by a ray determined by the Perron vector of a matrix related to the branching mechanism of the branching process in question.
Considering a critical branching random walk on the real line. From a study of the law of the trajectory of a particle chosen under the polymer measure, we establish a first order transition for the partition function at the critical parameter. This result is strongly related to a recent paper of Aïdékon and Shi in which they solved the problem of the normalisation of the partition function in the critical regime.
We obtain sharp asymptotic estimates for hitting probabilities of a critical branching Brownian motion in one dimension with killing at 0 We also obtain sharp asymptotic formulas for the tail probabilities of the number of particles killed at 0. In the special case of double-or-nothing branching, we give exact formulas for both the hitting probabilities, in terms of elliptic functions, and the distribution of the number of killed particles.
We establish recurrence and transience criteria for critical branching processes in random environment with immigration. These results are then applied to discuss recurrence and transience of a recurrent random walk in a random environment on Z that will be disturbed by cookies inducing a drift to the right of strength 1.
Using a high performance computer cluster, we run simulations regarding an open problem about d-dimensional critical branching random walks in a random IID environment The environment is given by the rule that at every site independently, with probability p>0, there is a cookie, completely suppressing the branching of any particle located there. Abstract. The simulations suggest self averaging: the asymptotic survival probability in n steps is the same in the annealed and the quenched case; it is \frac{2}{qn}, where q:=1-p. This particular asymptotics indicates a non-trivial phenomenon: the ta…
We investigate the long-time evolution of branching diffusion processes (starting with a single particle) in inhomogeneous media. The qualitative behavior of the processes depends on the intensity of the branching. We analyze the super-critical case, when the total number of particles growing exponentially with positive probability. We study the asymptotics of the number of particles in different regions of space and describe the growth of the region occupied by the particles.
Limit theorems are presented for the rescaled occupation time fluctuation process of a critical finite variance branching particle system in $\mathbb{R}^{d}$ with symmetric $α$-stable motion starting off from either a standard Poisson random field or the equilibrium distribution for critical $d=2α$ and large $d>2α$ dimensions. The limit processes are generalised Wiener processes. The obtained convergence is in space-time, finite-dimensional distributions sense. With the addtional assumption on the branching law we obtain functional convergence.
We consider a branching random walk on $\mathbb{R}$ with a killing barrier at zero. At criticality, the process becomes eventually extinct, and the total progeny $Z$ is therefore finite. We show that the tail distribution of $Z$ displays a typical behaviour in $(n\ln^2(n))^{-1}$, which confirms the prediction of Addario-Berry and Broutin.
The objective of this paper is the study of the equilibrium behavior of a population on the hierarchical group $Ω_N$ consisting of families of individuals undergoing critical branching random walk and in addition these families also develop according to a critical branching process. Strong transience of the random walk guarantees existence of an equilibrium for this two-level branching system. In the limit $N\to\infty$ (called the hierarchical mean field limit), the equilibrium aggregated populations in a nested sequence of balls $B^{(N)}_\ell$ of hierarchical radius $\ell$ converge to a backw…
We introduce and study the dynamics of an \emph{immortal} critical branching process. In the classic, critical branching process, particles give birth to a single offspring or die at the same rates. Even though the average population is constant in time, the ultimate fate of the population is extinction. We augment this branching process with immortality by positing that either: (a) a single particle cannot die, or (b) there exists an immortal stem cell that gives birth to ordinary cells that can subsequently undergo critical branching. We discuss the new dynamical aspects of this immortal bra…
This paper demonstrates a new regeneration processes technology making use of positive stable distributions. We study the asymptotic behavior of branching processes with a randomly controlled migration component. Using the new method, we confirm some known results and establish new limit theorems that hold in a more general setting.
In this paper, we investigate the asymptotic behaviors of the critical branching process with immigration $\{Z_n, n\ge 0\}$. First we get some estimation for the probability generating function of $Z_n$. Based on it, we get a large deviation for $Z_{n+1}/Z_n$. Lower and upper deviations for $Z_n$ are also studied. As a by-product, an upper deviation for $\max_{1\le i\le n} Z_i$ is obtained.
Let $\{Z_n\}_{n\geq 0 }$ be a critical or subcritical $d$-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on $\mathbb{R}^d$. Denote by $R_n:=\sup\{u>0:Z_n(\{x\in\mathbb{R}^d:|x|<u\})=0\}$ the radius of the largest empty ball centered at the origin of $Z_n$. In this work, we prove that after suitable renormalization, $R_n$ converges in law to some non-degenerate distribution as $n\to\infty$. Furthermore, our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random wal…
Using Foster-Lyapunov techniques we establish new conditions on non-extinction, non-explosion, coming down from infinity and staying infinite, respectively, for the general continuous-state nonlinear branching processes introduced in Li et al. (2019). These results can be applied to identify boundary behaviors for the critical cases of the above nonlinear branching processes with power rate functions driven by Brownian motion and (or) stable Poisson random measure, which was left open in Li et al. (2019). In particular, we show that even in the critical cases, a phase transition happens betwee…
We consider a critical continuous-time branching process (a Yule process) in which the individuals independently execute symmetric $α-$stable random motions on the real line starting at their birth points. Because the branching process is critical, it will eventually die out, and so there is a well-defined maximal location $M$ ever visited by an individual particle of the process. We prove that the distribution of $M$ satisfies the asymptotic relation $P\{M\geq x \}\sim (2/α)^{1/2}x^{-α/2}$ as $x \rightarrow \infty$.
We consider branching Brownian motion on the real line with absorption at zero, in which particles move according to independent Brownian motions with the critical drift of $-\sqrt{2}$. Kesten (1978) showed that almost surely this process eventually dies out. Here we obtain upper and lower bounds on the probability that the process survives until some large time $t$. These bounds improve upon results of Kesten (1978), and partially confirm nonrigorous predictions of Derrida and Simon (2007).
Let $(Z_n)_{n\geqslant 0}$ be a branching process in a random environment defined by a Markov chain $(X_n)_{n\geqslant 0}$ with values in a finite state space $\mathbb X$ starting at $X_0=i \in\mathbb X$. We extend from the i.i.d. environment to the Markovian one the classical classification of the branching processes into critical and strongly, intermediate and weakly subcritical states. In all these cases, we study the asymptotic behaviour of the probability that $Z_n>0$ as $n\to+\infty$.
We investigate the number of topological ends of the trace of branching random walk (BRW) on a graph, giving a sufficient condition for the trace to have infinitely many ends. We then describe some interesting examples of non-symmetric BRWs with just one end.
We consider an indecomposable Galton-Watson branching process with countably infinitely many types. Assuming that the process is critical and allowing for infinite variance of the offspring sizes of some (or all) types of particles we describe the asymptotic behavior of the survival probability of the process and establish a Yaglom-type conditional limit theorem for the infinite-dimensional vector of the number of particles of all types.
We focus on the existence and its characterization of limit for a certain critical branching random walks in time-space random environment in 1 dimension which was introduced by Birkner et.al. Each particle performs simple random walk on $\mathbb{Z}$ and branching mechanism depends on the time-space site. The weak limit points of this measure valued processes are characterized as a solution of the non-trivial martingale problem and called super-Brownian motions in random environment by Mytnik.
In this work, we characterize cluster-invariant point processes for critical branching spatial processes on R d for all large enough d when the motion law is $α$-stable or has a finite discrete range. More precisely, when the motion is $α$-stable with $α$ $\le$ 2 and the offspring law $μ$ of the branching process has an heavy tail such that $μ$(k) $\sim$ k --2--$β$ , then we need the dimension d to be strictly larger than the critical dimension $α$/$β$. In particular, when the motion is Brownian and the offspring law $μ$ has a second moment, this critical dimension is 2. Contrary to the previo…
We consider a branching Brownian motion with linear drift in which particles are killed on exiting the interval (0,K) and study the evolution of the process on the event of survival as the width of the interval shrinks to the critical value at which survival is no longer possible. We combine spine techniques and a backbone decomposition to obtain exact asymptotics for the near-critical survival probability. This allows us to deduce the existence of a quasi-stationary limit result for the process conditioned on survival which reveals that the backbone thins down to a spine as we approach critic…
In this paper we study the conditional limit theorems for critical continuous-state branching processes with branching mechanism $ψ(λ)=λ^{1+α}L(1/λ)$ where $α\in [0,1]$ and $L$ is slowly varying at $\infty$. We prove that if $α\in (0,1]$, there are norming constants $Q_{t}\to 0$ (as $t\uparrow +\infty$) such that for every $x>0$, $P_{x}\left(Q_{t}X_{t}\in\cdot|X_{t}>0\right)$ converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of $ψ$ at 0. We give a conditional limit theorem for the case $α=0$. The limit theorems we obtain in this paper allow…
The occupation time of an age-dependent branching particle system in $\Rd$ is considered, where the initial population is a Poisson random field and the particles are subject to symmetric $α$-stable migration, critical binary branching and random lifetimes. Two regimes of lifetime distributions are considered: lifetimes with finite mean and lifetimes belonging to the normal domain of attraction of a $γ$-stable law, $γ\in(0,1)$. It is shown that in dimensions $d>αγ$ for the heavy-tailed lifetimes case, and $d>α$ for finite mean lifetimes, the occupation time proccess satisfies a strong law of l…
We consider a branching random walk with an absorbing barrier, where the step of the associated one-dimensional random walk is in the domain of attraction of an $α$-stable law with $1<α<2$. We shall prove that there is a barrier $an^{\frac{1}{1+α}}$ and a critical value $a_α$ such that if $a<a_α$, then the process dies; if $a>a_α$, then the process survives. The results generalize previous results in literature for the case $α=2$.
We consider critical branching Brownian motion with absorption, in which there is initially a single particle at $x > 0$, particles move according to independent one-dimensional Brownian motions with the critical drift of $-\sqrt{2}$, and particles are absorbed when they reach zero. Here we obtain asymptotic results concerning the behavior of the process before the extinction time, as the position $x$ of the initial particle tends to infinity. We estimate the number of particles in the system at a given time and the position of the right-most particle. We also obtain asymptotic results for the…
We study the speed of extinction of continuous state branching processes in a Lévy environment, where the associated Lévy process oscillates. Assuming that the Lévy process satisfies the Spitzer's condition and the existence of some exponential moments, we extend recent results where the associated branching mechanism was stable. Our study relies on the path analysis of the process together with its environment, when this latter is conditioned to have a non negative running infimum. This approach is inspired from the discrete setting with i.i.d. environment studied in (Afanasyev et al. 2005).
In this paper the asymptotic behavior of a critical multi-type branching process with immigration is described when the offspring mean matrix is irreducible, in other words, when the process is indecomposable. It is proved that sequences of appropriately scaled random step functions formed from periodic subsequences of a critical indecomposable multi-type branching process with immigration converge weakly towards a process supported by a ray determined by the Perron vector of the offspring mean matrix. The types can be partitioned into nonempty mutually disjoint subsets (according to communica…
We study the local convergence of critical Galton-Watson trees and Levy trees under various conditionings. Assuming a very general monotonicity property on the functional of random trees, we show that random trees conditioned to have large functional values always converge locally to immortal trees. We also derive a very general ratio limit property for functionals of random trees satisfying the monotonicity property. Then we move on to study the local convergence of critical continuous-state branching processes, and prove a similar result. Finally we give a definition of continuum condensatio…
We are interested in the survival probability of a population modeled by a critical branching process in an i.i.d. random environment. We assume that the random walk associated with the branching process is oscillating and satisfies a Spitzer condition $\mathbf{P}(S_{n}>0)\rightarrow ρ,\ n\rightarrow \infty $, which is a standard condition in fluctuation theory of random walks. Unlike the previously studied case $ρ\in (0,1)$, we investigate the case where the offspring distribution is in the domain of attraction of a stable law with parameter $1$, which implies that $ρ=0$ or $1$. We find the a…
A Galton-Watson branching process with immigration evolving in a random environment is considered. Its associated random walk is assumed to be oscillating. We prove a functional limit theorem in which the process under consideration is normalized by a random coefficient depending on the random environment only. The distribution of the limiting process is described in terms of a strictly stable Levy process and a sequence of independent and identically distributed random variables which is independent of this process.
A critical branching process $\left\{ Z_{k},k=0,1,2,...\right\} $ in a random environment is considered. A conditional functional limit theorem for the properly scaled process $\left\{ \log Z_{pu},0\leq u<\infty \right\} $ is established under the assumptions $Z_{n}>0$ and $p\ll n$. It is shown that the limiting process is a Levy process conditioned to stay nonnegative. The proof of this result is based on a limit theorem describing the distribution of the initial part of the trajectories of a driftless random walk conditioned to stay nonnegative.
The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter $λ$. There is a threshold for $λ$, which is called $λ_w$, that separates almost sure global extinction from global survival. Analogously, there exists another threshold $λ_s$ below which any site is visited almost surely a finite number of times (i.e.~local extinction) while above it there is a positive probability of visiting every site infinitely many times. The local critical parameter $λ_s$ is completely understood and can be computed as a function of the reproduction rates. On the oth…
The asymptotic normality of conditional least squares estimators for the offspring variance in critical branching processes with non-homogeneous immigration is established, under moment assumptions on both reproduction and immigration. The proofs use martingale techniques and weak convergence results in Skorokhod spaces.
