2020/01/22 by Rytova, Anastasiya, Yarovaya, Elena
#60F99 #60G50 #60J80 #81Q15 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2001.08032
We study a continuous-time branching random walk on the lattice ℤd, d∈ ℕ, with a single source of branching, that is the lattice point where the birth and death of particles can occur. The random walk is assumed to be homogeneous, symmetric and irreducible but, in contrast to previous investigations, the random walk transition intensities a(x,y) decrease as |y-x|-(d+α) for |y-x|→ ∞, where α∈(0,2), that leads to an infinite variance of the random walk jumps. The~mechanism of the birth and death of particles at the source is governed by a continuous-time Bienaymé-Galton-Watson branching process. The source intensity is characterized by a certain parameter β. We calculate the long-time asymptotic behaviour for all integer moments for the number of particles at each lattice point and for the total population size. With respect to the parameter β a non-trivial critical point βc>0 is found for every d≥ 1. In particular, if β>βc the evolutionary operator generated a behaviour of the first moment for the number of particles has a positive eigenvalue. The existence of a positive eigenvalue yields an exponential growth in t of the particle numbers in the case β>βc called supercritical. Classification of the branching random walk treated as subcritical (β