Criteria of Exponential Growth for the Numbers of Particles in Models of Branching Random Walks

Abstract

We consider two models of continuous-time branching random walk on ${\bf Z}^d$ with one source of branching. The first model assumes that the random walk is symmetric over the entire lattice. In the second model, an additional parameter is introduced for “artificial” strengthening of the degree of dominance of branching or walk at the source, which violates the walk symmetry. Necessary and sufficient conditions are established under which the exponential growth in the numbers of particles is observed in the models both at an arbitrary lattice point and on the entire lattice. General methods for studying the models in the supercritical case are proposed.