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Quasi-stationary regime of a branching random walk in presence of an absorbing wall  [PDF]
Damien Simon,Bernard Derrida
Quantitative Biology , 2007, DOI: 10.1007/s10955-008-9504-4
Abstract: A branching random walk in presence of an absorbing wall moving at a constant velocity $v$ undergoes a phase transition as the velocity $v$ of the wall varies. Below the critical velocity $v_c$, the population has a non-zero survival probability and when the population survives its size grows exponentially. We investigate the histories of the population conditioned on having a single survivor at some final time $T$. We study the quasi-stationary regime for $v
Survival probability of the branching random walk killed below a linear boundary  [PDF]
Jean Bérard,Jean-Baptiste Gouéré
Mathematics , 2009,
Abstract: We give an alternative proof of a result by N. Gantert, Y. Hu and Z. Shi on the asymptotic behavior of the survival probability of the branching random walk killed below a linear boundary, in the special case of deterministic binary branching and bounded random walk steps. Connections with the Brunet-Derrida theory of stochastic fronts are discussed.
Asymptotics for the survival probability in a killed branching random walk  [PDF]
Nina Gantert,Yueyun Hu,Zhan Shi
Mathematics , 2008,
Abstract: Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope $\gamma-\epsilon$, where $\gamma$ denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when $\epsilon\to 0$, the probability in question decays like $\exp\{- {\beta + o(1)\over \epsilon^{1/2}}\}$, where $\beta$ is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli$(p)$ random variables (with $0
An upper bound for the probability of visiting a distant point by critical branching random walk in $\mathbb{Z}^4$  [PDF]
Qingsan Zhu
Mathematics , 2015,
Abstract: In this paper, we solve an open question raised by Le Gall and Lin. 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.
The Seneta--Heyde scaling for the branching random walk  [PDF]
Elie Aidekon,Zhan Shi
Mathematics , 2011, DOI: 10.1214/12-AOP809
Abstract: We consider the boundary case (in the sense of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609--631] in a one-dimensional super-critical branching random walk, and study the additive martingale $(W_n)$. We prove that, upon the system's survival, $n^{1/2}W_n$ converges in probability, but not almost surely, to a positive limit. The limit is identified as a constant multiple of the almost sure limit, discovered by Biggins and Kyprianou [Adv. in Appl. Probab. 36 (2004) 544--581], of the derivative martingale.
General discrete random walk with variable absorbing probabilities  [PDF]
Theo van Uem
Mathematics , 2009,
Abstract: We obtain expected number of arrivals, probability of arrival, absorption probabilities and expected time before absorption for a general discrete random walk with variable absorbing probabilities on a finite interval using Fibonacci numbers
Survival and Growth of a Branching Random Walk in Random Environment  [PDF]
Christian Bartsch,Nina Gantert,Michael Kochler
Mathematics , 2009,
Abstract: We consider a particular Branching Random Walk in Random Environment (BRWRE) on $\sN_0$ started with one particle at the origin. Particles reproduce according to an offspring distribution (which depends on the location) and move either one step to the right (with a probability in $(0,1]$ which also depends on the location) or stay in the same place. We give criteria for local and global survival and show that global survival is equivalent to exponential growth of the moments. Further, on the event of survival the number of particles grows almost surely exponentially fast with the same growth rate as the moments.
Coexistence in locally regulated competing populations and survival of branching annihilating random walk  [PDF]
Jochen Blath,Alison Etheridge,Mark Meredith
Mathematics , 2007, DOI: 10.1214/105051607000000267
Abstract: We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic results, and interacting particle system models, which do not, at present, incorporate all of the competitive strategies that a population might adopt. The second is a simplification of the first, in which competition is only supposed to act within lattice sites and the total population size within each lattice point is a constant. In a special case, this second model is dual to a branching annihilating random walk. For each model, using a comparison with oriented percolation, we show that for certain parameter values, both populations will coexist for all time with positive probability. As a corollary, we deduce survival for all time of branching annihilating random walk for sufficiently large branching rates. We also present a number of conjectures relating to the r\^{o}le of space in the survival probabilities for the two populations.
The critical random barrier for the survival of branching random walk with absorption  [PDF]
Bruno Jaffuel
Mathematics , 2009,
Abstract: We study a branching random walk on $\r$ with an absorbing barrier. The position of the barrier depends on the generation. In each generation, only the individuals born below the barrier survive and reproduce. Given a reproduction law, Biggins et al. \cite{BLSW91} determined whether a linear barrier allows the process to survive. In this paper, we refine their result: in the boundary case in which the speed of the barrier matches the speed of the minimal position of a particle in a given generation, we add a second order term $a n^{1/3}$ to the position of the barrier for the $n^\mathrm{th}$ generation and find an explicit critical value $a_c$ such that the process dies when $aa_c$. We also obtain the rate of extinction when $a < a_c$ and a lower bound on the surviving population when $a > a_c$.
Intermittency for branching random walk in heavy tailed environment  [PDF]
Marcel Ortgiese,Matthew I. Roberts
Mathematics , 2014,
Abstract: We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We describe the process, including a detailed shape theorem, in terms of a system of growing lilypads. As an application we show that the branching random walk is intermittent, in the sense that most particles are concentrated on one very small island with large potential. Moreover, we compare the branching random walk to the parabolic Anderson model and observe that although the two systems show similarities, the mechanisms that control the growth are fundamentally different.
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