Spatial structure of branching random walks

We consider time-continuous branching random walks on multidimensional lattices.
In the first part, the asymptotic results depending on the lattice dimension for symmetric branching random walks with a single source of branching and finite variance of jumps are presented. In particular asymptotic behavior of survival probabilities and limit theorems for the numbers of particles, both at an arbitrary point of the lattice and on the entire lattice, are obtained. After that we discuss the effects for branching random walks in another case when corresponding transition rates of the random walk have heavy tails. As a result, the variance of the jumps is infinite, and a random walk may be transient even on one- or two-dimensional lattices. Conditions of transience for a random walk and limit theorems for the numbers of particles are discussed.
In the second part, we present the results about branching random walks with violation of symmetry of the random walk at a source. Moreover, we introduce the general model of a branching random walk with a finitely many branching sources. In a supercritical case for such processes the phase transitions are discovered. This situation differs substantially from the branching random walks with a single source.
In the third part, the behavior of transition probabilities of a branching random walk in the situation when the space and time variables grow jointly are established. One of the main results here is the limit theorem about limiting properties of the Green function for the transition probabilities. These results are important for the investigation of the large deviations for branching random walks, in particular, for studying of the particle population front.
In the last part, we compare two models of homogeneous and non homogeneous branching random walks in random environments. In these models, the branching environment are formed of the birth-and-death processes at lattice sites with random intensities. Conditions under which the long-time behavior of the moments averaged by the medium coincide for the both models are obtained. It is shown that these assumptions hold for random potentials having Weibull-type and Gumbel-type upper tails.