For a continuous-time catalytic branching random walk (CBRW) on Z, with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include the regular variation of the jump distribution tail for underlying random walk and the well-known L log L condition for the offspring numbers. In our classification, given in the previous paper, the analysis refers to supercritical CBRW. The principle result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to a non-trivial law. An explicit formula is provided for this normalization and non-linear integral equations are obtained to determine the limiting distribution function. The novelty consists in establishing the weak convergence for CBRW with "heavy" tails, in contrast to the known behavior in case of "light" tails of the random walk jumps. The new tools such as "many-to-few lemma" and spinal decomposition appear non-efficient here. The approach developed in the paper combines the techniques of renewal theory, Laplace transform, non-linear integral equations and large deviations theory for random sums of random variables.
Keywords and phrases: catalytic branching random walk, heavy tails, regular varying tails, spread of population, L log L condition.
We consider a model of Branching Brownian Motion in which the usual spatially-homogeneous and catalytic branching at a single point are simultaneously present. We establish the almost sure growth rates of population in certain time-dependent regions and as a consequence the first-order asymptotic behaviour of the rightmost particle.
In this article we establish the magnitude of fluctuations of the extreme particle in the model of binary branching Brownian motion with a single catalytic point at the origin.
For many stochastic diffusion processes with mean field interaction, convergence of the rescaled total mass processes towards a diffusion process is known. Here we show convergence of the so-called finite system scheme for interacting jump-type processes known as mutually catalytic branching processes with infinite branching rate. Due to the lack of second moments the rescaling of time is different from the finite rate mutually catalytic case. The limit of rescaled total mass processes is identified as the finite rate mutually catalytic branching diffusion. The convergence of rescaled processe…
We consider a branching Brownian motion in which binary fission takes place only when particles are at the origin at a rate β> 0 on the local time scale. We obtain results regarding the asymptotic behaviour of the number of particles above λt at time t, for λ> 0. As a corollary, we establish the almost sure asymptotic speed of the rightmost particle. We also prove a Strong Law of Large Numbers for this catalytic branching Brownian motion.
We consider the model of branching Brownian motion with a single catalytic point at the origin and binary branching. We establish some fine results for the asymptotic behaviour of the numbers of particles travelling at different speeds and give an explicit characterisation of the spatial distribution of particles travelling at the critical speed.
We study the model of binary branching Brownian motion with spatially-inhomogeneous branching rate $βδ_0(\cdot)$, where $δ_0(\cdot)$ is the Dirac delta function and $β$ is some positive constant. We show that the distribution of the rightmost particle centred about $\fracβ{2}t$ converges to a mixture of Gumbel distributions according to a martingale limit. Our results form a natural extension to S. Lalley and T. Sellke [6] for the degenerate case of catalytic branching.
In this article we contribute to the moment analysis of branching processes in catalytic media. The many-to-few lemma based on the spine technique is used to derive a system of (discrete space) partial differential equations for the number of particles in a variation of constants formulation. The long-time behaviour is then deduced from renewal theorems and induction.
We construct a mutually catalytic branching process on a countable site space with infinite "branching rate". The finite rate mutually catalytic model, in which the rate of branching of one population at a site is proportional to the mass of the other population at that site, was introduced by Dawson and Perkins in [DP98]. We show that our model is the limit for a class of models and in particular for the Dawson-Perkins model as the rate of branching goes to infinity. Our process is characterized as the unique solution to a martingale problem. We also give a characterization of the process as …
We study the distribution of the maximal displacement of particles positions for the whole time of the population existence in the model of critical and subcritical catalytic branching random walk on Z. In particular, we prove that in the case of simple symmetric random walk on Z, the distribution of the maximal displacement has "a heavy tail" decreasing as a function of the power 1/2 or 1, when the branching process is critical or subcritical, respectively. These statements describe new effects which do not arise in the corresponding investigations of the maximal displacement of critical and …
Consider the infinite rate mutually catalytic branching process (IMUB) constructed in [Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence (2008) Preprint] and [Ann. Probab. 38 (2010) 479-497]. For finite initial conditions, we show that only one type survives in the long run if the interaction kernel is recurrent. On the other hand, under a slightly stronger condition than transience, we show that both types can coexist.
We consider a random walk on $\Z$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_n$. Then we determine all possible limiting law for the sequence $M_n -αn$ where $α$ is a deterministic constant.
Since the seminal work of Dawson and Perkins, mutually catalytic versions of super-processes have been studied frequently. In this article we combine two approaches extending their ideas: the approach of adding correlations to the driving noise of the system is combined with the approach of obtaining new processes by letting the branching rate tend to infinity. We introduce infinite rate symbiotic branching processes which surprisingly can be interpreted as generalized voter processes with additional strength of opinions. Since many of the arguments go along the lines of known proofs this arti…
We consider catalytic branching populations. They consist of a catalyst population evolving according to a critical binary branching process in continuous time with a constant branching rate and a reactant population with a branching rate proportional to the number of catalyst individuals alive. The reactant forms a process in random medium. We describe asymptotically the genealogy of catalytic branching populations coded as the induced forest of $\mathbb{R}$-trees using the many individuals--rapid branching continuum limit. The limiting continuum genealogical forests are then studied in detai…
In several examples, dualities for interacting diffusion and particle systems permit the study of the longtime behavior of solutions. A particularly difficult model in which many techniques collapse is a two-type model with mutually catalytic interaction introduced by Dawson/Perkins for which they proved under some assumptions a dichotomy between extinction and coexistence directly from the defining equations. In the present article we show how to prove a precise dichotomy for a related model with negatively correlated noises. The proof uses moment bounds on exit-times of correlated Brownian m…
In this paper, we study the weak convergence of the extremes of supercritical branching Lévy processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are Lévy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, $\mathbb{X}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of $\mathbb{X}_t$.
Consider the mutually catalytic branching process with finite branching rate $γ$. We show that as $γ\to\infty$, this process converges in finite-dimensional distributions (in time) to a certain discontinuous process. We give descriptions of this process in terms of its semigroup in terms of the infinitesimal generator and as the solution of a martingale problem. We also give a strong construction in terms of a planar Brownian motion from which we infer a path property of the process. This is the first paper in a series or three, wherein we also construct an interacting version of this process …
We study a catalytic branching process (CBP) with any finite set of catalysts. This model describes a system of particles where the movement is governed by a Markov chain with arbitrary finite or countable state space and the branching may only occur at the points of catalysis. The results obtained generalize and strengthen those known in cases of CBP with a single catalyst and of branching random walk on d-dimensional integer lattice with a finite number of sources of particles generation. We propose to classify CBP with N catalysts as supercritical, critical or subcritical according to the v…
For a supercritical catalytic branching random walk on Z^d (d is positive integer) with an arbitrary finite catalysts set we study the spread of particles population as time grows to infinity. Namely, we divide by t the position coordinates of each particle existing at time t and then let t tend to infinity. It is shown that in the limit there are a.s. no particles outside the closed convex surface in R^d which we call the propagation front and, under condition of infinite number of visits of the catalysts set, a.s. there exist particles on the propagation front. We also demonstrate that the p…
Subcritical catalytic branching random walk on d-dimensional lattice is studied. New theorems concerning the asymptotic behavior of distributions of local particles numbers are established. To prove the results different approaches are used including the connection between fractional moments of random variables and fractional derivatives of their Laplace transforms. In the previous papers on this subject only supercritical and critical regimes were investigated assuming finiteness of the first moment of offspring number and finiteness of the variance of offspring number, respectively. In the p…
Dawson and Perkins [Ann. Probab. 26 (1988) 1088--1138] constructed a stochastic model of an interacting two-type population indexed by a countable site space which locally undergoes a mutually catalytic branching mechanism. In Klenke and Mytnik [Preprint (2008), arXiv:0901.0623], it is shown that as the branching rate approaches infinity, the process converges to a process that is called the infinite rate mutually catalytic branching process (IMUB). It is most conveniently characterized as the solution of a certain martingale problem. While in the latter reference, a noise equation approach is…
We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten-Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the n-th generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of Brunet and Derrida (2011) remains valid in this setup, investigate various other issues mentioned in their paper and recover the main resu…
In this article, we consider a branching random walk on the real-line where displacements coming from the same parent have jointly regularly varying tails. The genealogical structure is assumed to be a supercritical Galton-Watson tree, satisfying Kesten-Stigum condition. We study the large deviations of the extremal process, formed by the appropriately normalized positions in the $n$-th generation and show that the large extreme-positions form clusters in the limit. As a consequence of this, we also study the large deviations of the maximum among positions at the $n$-th generation.
We study an iterated temporal and contemporaneous aggregation of $N$ independent copies of a strongly stationary subcritical Galton-Watson branching process with regularly varying immigration having index $α\in (0, 2)$. Limits of finite dimensional distributions of appropriately centered and scaled aggregated partial sum processes are shown to exist when first taking the limit as $N \to \infty$ and then the time scale $n \to \infty$. The limit process is an $α$-stable process if $α\in (0, 1) \cup (1, 2)$, and a deterministic line with slope $1$ if $α= 1$.
A catalytic branching random walk on a multidimensional lattice, with arbitrary finite number of catalysts, is studied in supercritical regime. The dynamics of spatial spread of the particles population is examined, upon normalization. The components of the vector random walk jump are assumed independent (or close to independent) and have semi-exponential distributions with, possibly, different parameters. A limit theorem on the almost sure normalized positions of the particles at the population ``front'' is established. Contrary to the case of the random walk increments with ``light'' distrib…
A general model of catalytic branching process (CBP) with any finite number of catalysis centers in a discrete space is studied. More exactly, it is assumed that particles move in this space according to a specified Markov chain and they may produce offspring only in the presence of catalysts located at fixed points. The asymptotic (in time) behavior of the total number of particles as well as the local particles numbers is investigated. The problems of finding the global extinction probability and local extinction probability are solved. Necessary and sufficient conditions are established for…
We consider a supercritical catalytic branching random walk (CBRW) on a multidimensional lattice Z^d (d is positive integer). The main subject of study is the behavior of particles cloud in space and time. For CBRW on an integer line, Carmona and Hu (2014) examined the asymptotical behavior of the maximal coordinate M_n of the particles at time n. They proved that M_n/n converges to μalmost surely (on a set of local non-degeneracy of CBRW), as n tends to infinity, where μ>0 is a certain constant. Under additional assumption of a single catalyst in CBRW they also investigated the fluctuations o…
In this article, we consider a Branching Random Walk on the real line. The genealogical structure is assumed to be given through a supercritical branching process in the i.i.d. environment and satisfies the Kesten-Stigum condition. The displacements coming from the same parent are assumed to have jointly regularly varying tails. Conditioned on the survival of the underlying genealogical tree, we prove that the appropriately normalized (normalization depends on the quenched size of the $n$-th generation) maximum among positions at the $n$-th generation converges weakly to a scale-mixture of Fre…
We study the extremes of branching random walks under the assumption that the underlying Galton-Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. We study the asymptotics of the scaled position of the rightmost particle in the $n$-th generation when the tail of the displacement behaves like $\exp(-K(x))$, where either $K$ is a regularly varying function of index $r> 0$, or $K$ …
We observe the continuous-time Markov Branching Process without high-order moments and allowing Immigration. Limit properties of transition functions and their convergence to invariant measures are investigated. Main mathematical tool is regularly varying generating functions with remainder.
We characterise all the quasi-stationary distributions and the Q-process associated with a continuous state branching process that explodes in finite time. We also provide a rescaling for the continuous state branching process conditioned on non-explosion when the branching mechanism is regularly varying at 0.
We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Markov branching process. We treat the critical case so that the infinitesimal generating function of the process has the infinite second moment, but it regularly varies with the remainder. We improve the Basic Lemma of the theory of critical Markov branching process and refine known limit results.
Critical catalytic branching random walk on d-dimensional integer lattice is investigated for all d. The branching may occur at the origin only and the start point is arbitrary. The asymptotic behavior, as time grows to infinity, is determined for the mean local particles numbers. The same problem is solved for the probability of particles presence at a fixed lattice point. Moreover, the Yaglom type limit theorem is established for the local number of particles. Our analysis involves construction of an auxiliary Bellman-Harris branching process with six types of particles. The proofs employ th…
Using the language of regular variation, we give a sufficient condition for a point process to be in the superposition domain of attraction of a strictly stable point process. This sufficient condition is then used to obtain an explicit representation of the weak limit of a sequence of point processes induced by a branching random walk with jointly regularly varying displacements. As a consequence, we extend the main result of Durrett (1983) and verify that two related predictions of Brunet and Derrida (2011) remain valid for this model.
Consider the continuous-time Markov Branching Process. In critical case we consider a situation when the generating function of intensity of transformation of particles has the infinite second moment, but its tail regularly varies in sense of Karamata. First we discuss limit properties of transition functions of the process. We prove local limit theorems and investigate ergodic properties of the process. Further we investigate limiting probability function conditioned to be never extinct. Hereupon we obtain a new stochastic population process as a continuous-time Markov chain called the Markov…
We consider a fixed-point equation for a non-negative integer-valued random variable, that appears in branching processes with state-independent immigration. A similar equation appears in the analysis of a single-server queue with a homogeneous Poisson input, feedback and permanent customer(s). It is known that the solution to this equation uniquely exists under mild first and logarithmic moments conditions. We find further the tail asymptotics of the distribution of the solution when the immigration size and branch size distributions are heavy-tailed. We assume that the distributions of inter…
We continue our study of the distribution of the maximal number $X^{\ast}_k$ of offsprings amongst all individuals in a critical Galton-Watson process started with $k$ ancestors, treating the case when the reproduction law has a regularly varying tail $\bar F$ with index $-α$ for $α>2$ (and hence finite variance). We show that $X^{\ast}_k$ suitably normalized converges in distribution to a Frechet law with shape parameter $α/2$; this contrasts sharply with the case $1<α<2$ when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in the dec…
In this paper we analyze a branching process with immigration defined recursively by $X_t=θ_t\circ X_{t-1}+B_t$ for a sequence $(B_t)$ of i.i.d. random variables and random mappings $ θ_t\circ x:=θ_t(x)=\sum_{i=1}^xA_i^{(t)}, $ with $(A_i^{(t)})_{i\in \mathbb{N}_0}$ being a sequence of $\mathbb{N}_0$-valued i.i.d. random variables independent of $B_t$. We assume that one of generic variables $A$ and $B$ has a regularly varying tail distribution. We identify the tail behaviour of the distribution of the stationary solution $X_t$. We also prove CLT for the partial sums that could be further gene…
We consider a branching random walk on a multi($Q$)-type, supercritical Galton-Watson tree which satisfies Kesten-Stigum condition. We assume that the displacements associated with the particles of type $Q$ have regularly varying tails of index $α$, while the other types of particles have lighter tails than that of particles of type $Q$. In this article, we derive the weak limit of the sequence of point processes associated with the positions of the particles in the $n^{th}$ generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process (SScDPP…
The $N$-particle branching random walk is a discrete time branching particle system with selection. We have $N$ particles located on the real line at all times. At every time step each particle is replaced by two offspring, and each offspring particle makes a jump of non-negative size from its parent's location, independently from the other jumps, according to a given jump distribution. Then only the $N$ rightmost particles survive; the other particles are removed from the system to keep the population size constant. Inspired by work of J. Bérard and P. Maillard, we examine the long term behav…
We provide a generalization of Theorem 1 in Bartkiewicz, Jakubowski, Mikosch and Wintenberger (2011) in the sense that we give sufficient conditions for weak convergence of finite dimensional distributions of the partial sum processes of a strongly stationary sequence to the corresponding finite dimensional distributions of a non-Gaussian stable process instead of weak convergence of the partial sums themselves to a non-Gaussian stable distribution. As an application, we describe the asymptotic behaviour of finite dimensional distributions of aggregation of independent copies of a strongly sta…
