Branching Random Walks with Heavy Tails

Abstract

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.

Keywords:

• Branching random walks
• Heavy tails
• Non homogeneous environments

Mathematics Subject Classification:

• Primary 60J80
• Secondary 60J35, 60J99

Acknowledgment

The author is grateful to Prof. S.A. Molchanov for the helpful ideas and discussions. The research is supported by RFBR grant 13-01-00653a.