Branching Random Walk Research Articles

A branching random walk in the presence of a hard wall

AbstractWe consider a branching random walk on a d-ary tree of height n ( $n \in \mathbb{N}$ ), in the presence of a hard wall which restricts each value to be positive, where d is a natural number satisfying $d\geqslant2$ . We consider the behaviour of Gaussian processes with long-range interactions, for example the discrete Gaussian free field, under the condition that it is positive on a large subset of vertices. We observe a relation with the expected maximum of the processes. We find the probability of the event that the branching random walk is positive at every vertex in the nth generation, and show that the conditional expectation of the Gaussian variable at a typical vertex, under positivity, is less than the expected maximum by order of $\log n$ .

Note on the Generalized Branching Random Walk on the Galton–Watson Tree

Let ∂T be a super-critical Galton–Watson tree. Recently, the first author computed almost surely and simultaneously the Hausdorff dimensions of the sets of infinite branches of the boundary of ∂T along which the sequence SnX(t)/SnX˜(t) has a given set of limit points, where SnX(t) and SnX˜(t) are two branching random walks defined on ∂T. In this study, we are interested in the study of the speed of convergence of this sequence. More precisely, for a given sequence s=(sn), we consider Eα,s=t∈∂T:SnX(t)−αSnX˜(t)∼snasn→+∞. We will give a sufficient condition on (sn) so that Eα,s has a maximal Hausdorff and packing dimension.

Large deviations of extremes in branching random walk with regularly varying displacements

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.

The maximum of a branching random walk with stretched exponential tails

Nous étudions une marche aléatoire branchante uni-dimensionelle quand les déplacements n’ont pas des moments exponentiels. Plus précisement, la queue d’un déplacement X se comporte comme suit : P[X>t]∼aexp{−λtr}, pour des constantes a,λ>0 et r∈(0,1). Nous donnons une description détaillée du comportement asymptotique du maximum, en montrant des lois limites presque sûres, des theorèmes de convergence en loi et une dichotomie basée sur une condition de croissance. Ces lois limites diverses font apparaître des différences interéssantes entre les deux régimes r∈(0,2/3) et r∈(2/3,1), et le cas critique r=2/3 est encore différent.

Free energy subadditivity for symmetric random Hamiltonians

We consider a random Hamiltonian H:Σ→R defined on a compact space Σ that admits a transitive action by a compact group G. When the law of H is G-invariant, we show its expected free energy relative to the unique G-invariant probability measure on Σ, which obeys a subadditivity property in the law of H itself. The bound is often tight for weak disorder and relates free energies at different temperatures when H is a Gaussian process. Many examples are discussed, including branching random walks, several spin glasses, random constraint satisfaction problems, and the random field Ising model. We also provide a generalization to quantum Hamiltonians with applications to the quantum Sherrington–Kirkpatrick and Sachdev–Ye–Kitaev models.

The Spread Speed of Multiple Catalytic Branching Random Walks

The Spread Speed of Multiple Catalytic Branching Random Walks

Branching Random Walks with One Particle Generation Center and Possible Absorption at Every Point

We consider a new model of a branching random walk on a multidimensional lattice with continuous time and one source of particle reproduction and death, as well as an infinite number of sources in which, in addition to the walk, only the absorption of particles can occur. The asymptotic behavior of the integer moments of both the total number of particles and the number of particles at a lattice point is studied depending on the relationship between the model parameters. In the case of the existence of an isolated positive eigenvalue of the evolution operator of the average number of particles, a limit theorem is obtained on the exponential growth of both the total number of particles and the number of particles at a lattice point.

Branching random walk with infinite progeny mean: A tale of two tails

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 nth 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 has an exponential growth. We identify the exact scaling of the maxima in all cases and show the existence of a non-trivial limit when r>1.

On the role of reduced habitat in the phase transition of a stochastic model for seed dispersal

Habitat loss is one of the biggest threats facing plant species nowadays. We formulate a simple mathematical model of seed dispersal on reduced habitats to discuss survival of the species in relation to the habitat size and seeds production rate. Seeds get dispersed around the mother plant via several agents in a random way. In our model, seeds landing sites are distributed according to a homogeneous Poisson point process with a constant rate on . We will assume that each seed will successfully germinate and grow into a new plant with the same characteristics as the mother plant. The time is discrete, scaled according to generations of plants, or can represent years, since annual plants go through an entire growing cycle during 1 year. Then we will assume there are two symmetric barriers with respect to the origin and consider that the growth cannot evolve past the barriers. Imposing barriers correspond to the physical limitation of the habitat. We appeal to tools of probability theory to formalize and study such a model, which can be seen as a discrete‐time one‐dimensional branching random walk with barriers. By means of coupling techniques and the comparison with suitably constructed multitype branching processes, we localize the critical parameter of the process around which there is survival with positive probability or extinction almost surely. In addition, we consider a discrete‐space version of the model for which exact results are also obtained.

Scaling limits and fluctuations of a family of N-urn branching processes

In this paper, we are concerned with a family of N-urn branching processes, where some particles are initially placed in N urns, and then each particle gives birth to several new particles in some urn when dies. This model includes the N-urn Ehrenfest model and N-urn branching random walk as special cases. We show that the scaling limit of the process is driven by a C(T)-valued linear ordinary differential equation and the fluctuation of the process is driven by a generalized Ornstein–Uhlenbeck process in the dual of C∞(T), where T=(0,1] is the one-dimensional torus. A crucial step for the proofs of the above main results is to show that numbers of particles in different urns are approximately independent. As applications of our main results, the limit theorems of the hitting times of the process are also discussed.

A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process

A continuous-time particle system on the real line satisfying the branching property and an exponential integrability condition is called a branching L\'evy process, and its law is characterized by a triplet $(\sigma^2,a,\Lambda)$. We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of $(\sigma^2,a,\Lambda)$. This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred L\'evy processes conditioned to stay positive.

Solutions of kinetic-type equations with perturbed collisions

We study a class of kinetic-type differential equations ∂ϕt/∂t+ϕt=Q̂ϕt, where Q̂ is an inhomogeneous smoothing transform and, for every t≥0, ϕt is the Fourier–Stieltjes transform of a probability measure. We show that under mild assumptions on Q̂ the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk pertaining to Q̂. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as t→∞.

Development and implementation of branching random walk on spheres algorithms for solving the 2D elastostatics Lamé equation

Abstract In this paper, we address a long-standing open problem in stochastic simulation: construction of a random walk on spheres (RWS) algorithm for solving a system of elasticity equations, known as the Lamé equation. Many attempts to generalize the classic probabilistic representations like the Kac formula for parabolic and scalar elliptic equations failed. A different approach based on a branching random walk on spheres (BRWS) introduced in our paper of 1995 [K. K. Sabelfeld and D. Talay, Integral formulation of the boundary value problems and the method of random walk on spheres, Monte Carlo Methods Appl. 1 1995, 1, 1–34] made little progress in solving this problem. In the present study, we further improve the BRWS algorithm by a special implementation of a branching anisotropic random walk on spheres process.

Moments, large and moderate deviations for branching random walks with immigration in random environments

We consider an Rd-valued branching random walk with immigration in an i.i.d. time-dependent environment. For t∈Rd, let Zn(t) be the partition function of the system, and Wn(t) be the intrinsic sub-martingale formed by the normalization of Zn(t). By decomposing the family tree and using sums to characterise the upper bounds of the moments of Biggins's martingale, we study the change rates of the moments EWn(t)s for s∈R, and give sufficient conditions for the finiteness of supn⁡EWn(t)s. Based on these moment results, large and moderate deviation principles are established for log⁡Zn(t).

Moderate deviation probabilities for empirical distribution of the branching random walk

Given a super-critical branching random walk {Zn}n≥0 on R, let Zn(A) be the number of particles located in some Borel set A⊂R at generation n. Under some mild conditions, it's well-known (e.g., [4]) that Zn(nA)/Zn(R) converges almost surely to ν(A) as n→∞, where ν(⋅) is the standard Gaussian measure on R. Large deviation probabilities of Zn(nA)/Zn(R) have been well studied (see [10] and [27]). In this work, we investigate its moderate deviation probabilities, i.e. the convergence rate ofP(Zn(nA)/Zn(R)≥ν(A)+Δn) as n→∞, where {Δn}n≥0 is some positive sequence tending to zero.

Asymptotic behavior of survival probability for a branching random walk with a barrier

<abstract><p>Consider a branching random walk with a mechanism of elimination. We assume that the underlying Galton-Watson process is supercritical, thus the branching random walk has a positive survival probability. A mechanism of elimination, which is called a barrier, is introduced to erase the particles who lie above $ ri+\varepsilon i^{\alpha} $ and all their descendants, where $ i $ presents the generation of the particles, $ \alpha > 1/3, \varepsilon\in{{\mathbb{R}}} $ and $ r $ is the asymptotic speed of the left-most position of the branching random walk. First we show that the particle system still has a positive survival probability after we introduce the barrier with $ \varepsilon > 0. $ Moreover, we show that the decay of the probability is faster than $ e^{-\beta'\varepsilon^{\beta}} $ as $ \varepsilon\downarrow 0 $, where $ \beta', \beta $ are two positive constants depending on the branching random walk and $ \alpha $. The result in the present paper extends a conclusion in Gantert et al. (2011) in some extent. Our proof also works for some time-inhomogeneous cases.</p></abstract>

Conditional central limit theorem for subcritical branching random walk

Conditional central limit theorem for subcritical branching random walk

On one limit theorem for branching random walks

Общая теория марковских случайных процессов была заложена А. Н. Колмогоровым. К таким процессам относятся ветвящиеся случайные блуждания по решеткам $\mathbf{Z}^d$, $d\in\mathbf{N}$. Мы рассмотрим ветвящееся случайное блуждание, в котором частицы могут погибать и производить потомство, находясь в любой точке решетки. Перемещение каждой частицы по $\mathbf{Z}^d$ описывается симметричным, однородным и неприводимым случайным блужданием. Интенсивность ветвления частиц в точке $x\in \mathbf{Z}^d$ стремится к нулю при $\|x\|\to\infty$, и выполнено дополнительное условие на параметры ветвящегося случайного блуждания, гарантирующее экспоненциальный по времени рост среднего числа частиц в каждой точке $\mathbf{Z}^d$. В правой части уравнения для среднего числа частиц в этом случае возникает возмущение оператора, задающего блуждание, вызванное возможностью генерации частиц в точке $\mathbf{Z}^d$. Подобного рода уравнения с возмущением оператора диффузии в $\mathbf{R}^2$ были рассмотрены в работе Колмогорова-Петровского-Пискунова в 1937 г. и продолжают исследоваться в теории ветвящихся случайных блужданий на дискретных структурах. В перечисленных выше предположениях доказывается предельная теорема о сходимости в среднеквадратическом нормированного числа частиц в произвольной фиксированной точке решетки при $t\to\infty$.

Invariant measures of critical branching random walks in high dimension

In this work, we characterize cluster-invariant point processes for critical branching spatial processes on Rd 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≤2 and the offspring law μ of the branching process has an heavy tail such that μ(k)∼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 previous work of Bramson, Cox and Greven in [4] whose proof used PDE techniques, our proof uses probabilistic tools only.

On the boundary at infinity for branching random walk

On the boundary at infinity for branching random walk