Maxima of branching random walks vs. independent random walks

In recent years several authors have obtained limit theorems for the location of the right most particle in a supercritical branching random walk. In this paper we will consider analogous problems for an exponentially growing number of independent random walks. A comparison of our results with the known results of branching random walk then identifies the limit behaviors which are due to the number of particles and those which are determined by the branching structure.     

Localization for branching random walks in random environment

We consider branching random walks in d$d$-dimensional integer lattice with time–space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If d≥3$d\ge 3$ and the environment is “not too random”, then, the total population grows as fast as its expectation with strictly positive probability. If, on the other hand, d≤2$d\le 2$, or the environment is “random enough”, then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of “replica overlap”. We also prove a certain stronger localization property, whenever the total population grows strictly slower than its expectation almost surely.     

Survival of branching random walks with absorption

We consider a branching random walk on R starting from x≥0 and with a killing barrier at 0. At each step, particles give birth to b children, which move independently. Particles that enter the negative half-line are killed. In the case of almost sure extinction, we find asymptotics for the survival probability at time n, when n tends to infinity.     

Anisotropic branching random walks on homogeneous trees

Symmetric branching random walk on a homogeneous tree exhibits a weak survival phase: For parameter values in a certain interval, the population survives forever with positive probability, but, with probability one, eventually vacates every finite subset of the tree. In this phase, particle trails must converge to the geometric boundaryΩ of the tree. The random subset Λ of the boundary consisting of all ends of the tree in which the population survives, called the limit set of the process, is shown to have Hausdorff dimension no larger than one half the Hausdorff dimension of the entire geometric boundary. Moreover, there is strict inequality at the phase separation point between weak and strong survival except when the branching random walk is isotropic. It is further shown that in all cases there is a distinguished probability measure μ supported by Ω such that the Hausdorff dimension of Λ∩Ω_μ, where Ω_μ is the set of μ-generic points of Ω, converges to one half the Hausdorff dimension of Ω_μ at the phase separation point. Exact formulas are obtained for the Hausdorff dimensions of Λ and Λ∩Ω_μ, and it is shown that the log Hausdorff dimension of Λ has critical exponent 1/2 at the phase separation point. Received: 30 June 1998 / Revised version: 10 March 1999     

Recurrence and transience of multitype branching random walks

We study a discrete time Markov process with particles being able to perform discrete time random walks and create new particles, known as branching random walk (BRW). We suppose that there are particles of different types, and the transition probabilities, as well as offspring distribution, depend on the type and the position of the particle. Criteria of (strong) recurrence and transience are presented, and some applications (spatially homogeneous case, Lamperti BRW, many-dimensional BRW) are studied.     

Limit distributions for minimal displacement of branching random walks

Summary We study the minimal displacement (X _n ) of branching random walk with non-negative steps. It is shown that (X _n –EX _n ) is tight under a mild moment condition on the displacements. For supercritical B.R.W. (X _n ) converges almost surely. For critical B.R.W. we determine the possible limit points of (X _n –EX _n ), and we prove a generalization of Kolmogorov's theorem on the extinction probability of a critical branching process. Finally we generalize Bramson's results on the almost sure convergence ofX _n log 2/log logn.     

Limiting point processes in the branching random walk

Summary Take the nth generation of a supercritical branching random walk (a spatially homogeneous branching process) as a process of cluster centres and take independent copies of some simple point process Y as the clusters. Let the resulting point process be Y _n . For a given sequence of real numbers {x _n } let Y _n be centred on x _n . Under certain conditions, when an appropriate scale change is made, the resulting point process converges in distribution to a non-trivial limit.     

Limit distributions arising in branching random walks on integer lattices

The paper completes the investigation of limit distribution of the number of particles at the source of branching in the model of critical catalytic branching random walk on ^dd N {{mathbb Z}^d};d in {mathbb N} . Limit theorems of such kind were established only for d = 1, 2, 3, 4 under the assumption that, at the initial moment, there is a single particle at the source of branching. We prove their analog for d geqslant 5 d geqslant 5 . Moreover, in any dimension, we generalize the previous results by permitting the initial particle to start at an arbitrary point of the lattice.     

Limit distributions for the number of particles in branching random walks

We study branching random walks with continuous time. Particles performing a random walk on ?², are allowed to be born and die only at the origin. It is assumed that the offspring reproduction law at the branching source is critical and the random walk outside the source is homogeneous and symmetric. Given particles at the origin, we prove a conditional limit theorem for the joint distribution of suitably normalized numbers of particles at the source and outside it as time unboundedly increases. As a consequence, we establish the asymptotic independence of such random variables.     

Asymptotic properties of supercritical branching processes in random environments

We consider a supercritical branching process （Zn） in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the natural martingale Wn = Zn/E[Zn｜ξ], the convergence rates of W - Wn （by considering the convergence in law with a suitable norming, the almost sure convergence, the convergence in Lp, and the convergence in probability）, and limit theorems （such as central limit theorems, moderate and large deviations principles） on （log Zn）.     

A weakly supercritical mode in a branching random walk

The case of weakly supercritical branching random walks is considered. A theorem on asymptotic behavior of the eigenvalue of the operator defining the process is obtained for this case. Analogues of the theorems on asymptotic behavior of the Green function under large deviations of a branching random walk and asymptotic behavior of the spread front of population of particles are established for the case of a simple symmetric branching random walk over a many-dimensional lattice. The constants for these theorems are exactly determined in terms of parameters of walking and branching.     

Limit theorems for supercritical branching random fields with immigration

A measure-valued process which carries genealogical information is defined for a supercritical branching random field with immigration. This process counts the particles present at a final time whose ancestors had specified locations at given times in the past. A law of large numbers and a fluctuation limit theorem are proved for this process under a space-time scaling. The fluctuation limit is a nonstationary generalized Ornstein-Uhlenbeck process. An example of interest in transport theory and polymer chemistry is given.     

Maxima of random properties of particles in supercritical branching processes

Maxima of i.i.d. random variables are considered in the case when the growth of a sample is described by a supercritical branching process without degeneration. Limit theorems on distributions of maxima are proved. Some examples are presented. A connection with maximum-semistable laws is revealed.     

Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment

The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this paper is to analyze a time-inhomogeneous model with two levels of randomness. In the first step a sequence of branching laws is sampled independently according to a distribution on the set of point measures’ laws. Conditionally on the realization of this sequence (called environment) we define a branching random walk and find the asymptotic behavior of its maximal particle. It is of the form $V_n?\phi logn+o_{\mathbf{P}}(logn)$, where $V_n$ is a function of the environment that behaves as a random walk and $\phi >0$ is a deterministic constant, which turns out to be bigger than the usual logarithmic correction of the homogeneous branching random walk.     

Lower large deviations for supercritical branching processes in random environment

Branching processes in random environment (Z _n : n ≥ 0) are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical regime, the process survives with a positive probability and grows exponentially on the non-extinction event. We focus on rare events when the process takes positive values but lower than expected. More precisely, we are interested in the lower large deviations of Z, which means the asymptotic behavior of the probability {1 ≤ Z _n ≤ exp(nθ)} as n → ∞. We provide an expression for the rate of decrease of this probability under some moment assumptions, which yields the rate function. With this result we generalize the lower large deviation theorem of Bansaye and Berestycki (2009) by considering processes where ?(Z ₁ = 0 | Z ₀ = 1) > 0 and also much weaker moment assumptions.     

Upper tails for intersection local times of random walks in supercritical dimensions

We determine the precise asymptotics of the logarithmic uppertail probability of the total intersection local time of p independentrandom walks in ^d under the assumption p(d–2) > d.Our approach allows a direct treatment of the infinite timehorizon.     

A class of infinitely divisible distributions connected to branching processes and random walks

A class of infinitely divisible distributions on {0,1,2,…} is defined by requiring the (discrete) Lévy function to be equal to the probability function except for a very simple factor. These distributions turn out to be special cases of the total offspring distributions in (sub)critical branching processes and can also be interpreted as first passage times in certain random walks. There are connections with Lambert's W function and generalized negative binomial convolutions.