Supercritical Branching Random Walks with a Single Source

Abstract

We consider two continuous-time branching random walks on multidimensional lattices with birth and death of particles at the origin. In the first one, the underlying random walk is assumed to be symmetric. In the second one, an additional parameter is introduced to intensify artificially the prevalence of branching or walk at the origin. As a side effect, it violates symmetry of the random walk. Necessary and sufficient conditions for exponential growth for the numbers of particles both at an arbitrary point of the lattice and on the entire lattice are obtained. General methods to study the models in the supercritical case are proposed.