Branching Random Walk in a Catalytic Medium. I. Basic Equations

We consider a continuous-time branching random walk on the integer lattice ^d (d 1 ) with a finite number of branching sources, or catalysts. The random walk is assumed to be spatially homogeneous and irreducible. The branching mechanism at each catalyst, being independent of the random walk, is governed by a Markov branching process. The quantities of interest are the local numbers of particles (at each site) and the total population size. In the present paper, we derive and analyze the Kolmogorov type backward equations for the corresponding Laplace generating functions and also for the successive integer moments and the process extinction probability. In particular, existence and uniqueness theorems are proved and the problem of explosion is studied in some detail. We then rewrite these equations in the form of integral equations of renewal type, which may serve as a convenient tool for the study of the process long-time behavior. The paper also provides a technical foundation to some results published before without detailed proofs.     

Catalytic branching random walks in ℤ d with branching at the origin

A time-continuous branching random walk on the lattice ? ^d , d ≥ 1, is considered when the particles may produce offspring at the origin only. We assume that the underlying Markov random walk is homogeneous and symmetric, the process is initiated at moment t = 0 by a single particle located at the origin, and the average number of offspring produced at the origin is such that the corresponding branching random walk is critical. The asymptotic behavior of the survival probability of such a process at moment t → ∞ and the presence of at least one particle at the origin is studied. In addition, we obtain the asymptotic expansions for the expectation of the number of particles at the origin and prove Yaglom-type conditional limit theorems for the number of particles located at the origin and beyond at moment t.     

Critical dimension for a model of branching in a random medium

Summary A critical branching random walk in a d-dimensional spatial random medium (environment) is considered. It is said to be persistent if there is no loss of particle intensity in the large time limit. A critical dimension d _c is shown to exist such that the system is persistent if d>d _c and fails to be persistent if d _c.Research supported by the Natural Sciences and Engineering Research Council of Canada     

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.     

A random walk with a branching system in random environments

We consider a branching random walk in random environments, where the particles are reproduced as a branching process with a random environment (in time), and move independently as a random walk on Z with a random environment (in locations). We obtain the asymptotic properties on the position of the rightmost particle at time n, revealing a phase transition phenomenon of the system.     

The monotonicity of the probability of return into the source in models of branching random walks

The paper discusses two models of a branching random walk on a many-dimensional lattice with birth and death of particles at a single node being the source of branching. The random walk in the first model is assumed to be symmetric. In the second model an additional parameter is introduced which enables “artificial” intensification of the prevalence of branching or walk at the source and, as the result, violating the symmetry of the random walk. The monotonicity of the return probability into the source is proved for the second model, which is a key property in the analysis of branching random walks.     

On total progeny of multitype Galton-Watson process and the first passage time of random walk on lattice

In this paper,we form a method to calculate the probability generating function of the total progeny of multitype branching process.As examples,we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and(2-1) random walk.Consequently,we could give the probability generating functions and the distributions of the first passage time of corresponding random walks.Especially,for recurrent random walk which could stay at its position with probability 0 r 1,we show that the tail probability of the first passage time decays as 2/(π(1-r)~(1/2)) n~(1/1)= when n →∞.     

Subcritical branching processes in a random environment without the Cramer condition

A subcritical branching process in random environment (BPRE) is considered whose associated random walk does not satisfy the Cramer condition. The asymptotics for the survival probability of the process is investigated, and a Yaglom type conditional limit theorem is proved for the number of particles up to moment $n$ given survival to this moment. Contrary to other types of subcritical BPRE, the limiting distribution is not discrete. We also show that the process survives for a long time owing to a single big jump of the associate random walk accompanied by a population explosion at the beginning of the process.     

A note on exact convergence rate in the local limit theorem for a lattice branching random walk

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton-Watson process and the moving law of particles by a discrete random variable on the integer lattice Z. Denote by Z_n(z) the number of particles in the n-th generation in the model for each z ∈ Z. We derive the exact convergence rate in the local limit theorem for Z_n(z) assuming a condition like "EN(log N)~(1+λ) ∞" for the offspring distribution and a finite moment condition on the motion law. This complements the known results for the strongly non-lattice branching random walk on the real line and for the simple symmetric branching random walk on the integer lattice.     

Tree-indexed random walks on groups and first passage percolation

Summary Suppose that i.i.d. random variables are attached to the edges of an infinite tree. When the tree is large enough, the partial sumsS along some of its infinite paths will exhibit behavior atypical for an ordinary random walk. This principle has appeared in works on branching random walks, first-passage percolation, and RWRE on trees. We establish further quantitative versions of this principle, which are applicable in these settings. In particular, different notions of speed for such a tree-indexed walk correspond to different dimension notions for trees. Finally, if the labeling variables take values in a group, then properties of the group (e.g., polynomial growth or a nontrivial Poisson boundary) are reflected in the sample-path behavior of the resulting tree-indexed walk.Partially supported by a grant from the Landau Center for Mathematical AnalysisPartially supported by NSF grant DMS-921 3595     

Branching random walk with trapping zones

We consider a branching random walk with values in a certain set $\mathbb{S}$, where the branching mechanism is different according to whether particles (individuals) are in a certain so called trapping set $A?\mathbb{S}$ or not. We are then interested, under different scenarios, in properties of either the transient random measure describing distribution of individuals on $\mathbb{S}$ over time or its asymptotic behaviour.     

On the Mean Number of Particles of a Branching Random Walk on ℤ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> with Periodic Sources of Branching

We consider a continuous-time branching random walk on ? ^d , where the particles are born and die on a periodic set of points (sources of branching). The spectral properties of the evolution operator for the mean number of particles at an arbitrary point of ? ^d are studied. This operator is proved to have a positive spectrum, which leads to an exponential asymptotic behavior of the mean number of particles as t → ∞.     

Coalescing and annihilating random walk with ‘action at a distance’

We consider a system of particles which perform continuous time random walks onZ ^d . These random walks are independent as long as no two particles are at the same site or adjacent to each other. When a particle jumps from a site x to a sitey and there is already another particle aty or at some neighbory′ ofy, then there is an interaction. In the coalescing model, either the particle which just jumped toy is removed (or, equivalently, coalesces with a particle aty ory′) or all the particles at the sites adjacent toy (other thanx) are removed. In the annihilating random walk, the particle which just jumped toy and one particle aty ory′ annihilate each other. We prove that when the dimensiond is at least 9, then the density of this system is asymptotically equivalent toC/t for some constant C, whose value is explicitly given.     

Local Particles Numbers in Critical Branching Random Walk

Critical catalytic branching random walk on an integer lattice ? ^d 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 the presence of particles 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 the asymptotic properties of the (improper) c.d.f. of hitting times with taboo. The latter notion was recently introduced by the author for a non-branching random walk on ? ^d .     

Total progeny in killed branching random walk

We consider a branching random walk for which the maximum position of a particle in the n??th generation, R _n , has zero speed on the linear scale: R _n /n ?? 0 as n ?? ??. We further remove (??kill??) any particle whose displacement is negative, together with its entire descendence. The size Z of the set of un-killed particles is almost surely finite (Gantert and Müller in Markov Process. Relat. Fields 12:805?C814, 2006; Hu and Shi in Ann. Probab. 37(2):742?C789, 2009). In this paper, we confirm a conjecture of Aldous (Algorithmica 22:388?C412, 1998; and Power laws and killed branching random walks) that E [Z]?<??? while ${{\mathbf E}\left[Z\,{\rm log}\, Z\right]=\infty}$ . The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks.     

Canonical ensemble of particles in a self-avoiding random walk

We consider an ensemble of particles not interacting with each other and randomly walking in the d-dimensional Euclidean space ? ^d . The individual moves of each particle are governed by the same distribution, but after the completion of each such move of a particle, its position in the medium is “marked” as a region in the form of a ball of diameter r ₀, which is not available for subsequent visits by this particle. As a result, we obtain the corresponding ensemble in ? ^d of marked trajectories in each of which the distance between the centers of any pair of these balls is greater than r ₀. We describe a method for computing the asymptotic form of the probability density W _n (r) of the distance r between the centers of the initial and final balls of a trajectory consisting of n individual moves of a particle of the ensemble. The number n, the trajectory modulus, is a random variable in this model in addition to the distance r. This makes it necessary to determine the distribution of n, for which we use the canonical distribution obtained from the most probable distribution of particles in the ensemble over the moduli of their trajectories. Averaging the density W _n (r) over the canonical distribution of the modulus n allows finding the asymptotic behavior of the probability density of the distance r between the ends of the paths of the canonical ensemble of particles in a self-avoiding random walk in ? ^d for 2 ≤ d < 4.     

A Monotonicity Result for the Range of a Perturbed Random Walk

We consider a discrete time simple symmetric random walk on \(\mathbb{Z }^d,\,d\ge 1,\) where the path of the walk is perturbed by inserting deterministic jumps. We show that for any time \(n\in \mathbb{N }\) and any deterministic jumps that we insert, the expected number of sites visited by the perturbed random walk up to time \(n\) is always larger than or equal to that for the unperturbed walk. This intriguing problem arises from the study of a particle among a Poisson system of moving traps with sub-diffusive trap motion. In particular, our result implies a variant of the Pascal principle, which asserts that among all deterministic trajectories the particle can follow, the constant trajectory maximizes the particle’s survival probability up to any time \(t>0.\)