Branching Random Walks with Heavy Tails

We consider a continuous-time branching random walk on Z d , where the particles are born and die at a single lattice point (the source of branching). The underlying random walk is assumed to be symmetric. Moreover, 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 low-dimensional lattices (d = 1, 2). Conditions of transience for a random walk on Z d and limit theorems for the numbers of particles both at an arbitrary point of the lattice and on the entire lattice are obtained.