Branching Random Walks with Various Space Dynamics and a Finite Set of Centers for Particle Generation

The results of the investigation of phase transitions for various continuous-time branching random walks (BRWs) on multidimensional integer lattices with a few sources of branching are presented. Models of BRWs with symmetric and nonsymmetric underlying random walks with finite variance of jumps are studied. Moreover, symmetric BRWs are considered under the assumption that the corresponding transition rates of the random walk are homogeneous by space and have heavy tails. Such BRWs possess an infinite variance of jumps and, as a result, the random walk may be transient even on low-dimensional lattices (d=1,2). Conditions of transience for a random walk on multidimensional lattices and limit theorems for the numbers of particles, both at an arbitrary point of the lattice and on the entire lattice, are obtained. To study the front of the particle population, the asymptotic behavior of transition probabilities and the resolvent of the generator of random walk is investigated in detail. These results will be applied to study of BRWs on multidimensional lattices for obtaining of new limit theorems.
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