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Supercritical branching random walks with a single source. (English) Zbl 1229.60100

Summary: 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.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F99 Limit theorems in probability theory
60J35 Transition functions, generators and resolvents
60G50 Sums of independent random variables; random walks
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