Heavy-Tailed Branching Random Walks on Multidimensional Lattices. A Moment Approach

Abstract: We study a continuous-time branching random walk on the lattice $\mathbb{Z}^{d}$, $d\in \mathbb{N}$, 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 $|y-x|^{-(d+\alpha)}$ for $|y-x|\to \infty$, where $\alpha\in(0,2)$, 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 Bienaym\'e-Galton-Watson branching process. The source intensity is characterized by a certain parameter $\beta$. 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 $\beta$ a non-trivial critical point $\beta_c>0$ is found for every $d\geq 1$. In particular, if $\beta>\beta_{c}$ 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 $\beta>\beta_c$ called \emph{supercritical}. Classification of the branching random walk treated as \emph{subcritical} ($\beta<\beta_c$) or \emph{critical} ($\beta=\beta_c$) for the heavy-tailed random walk jumps is more complicated than for a random walk with a finite variance of jumps. We study the asymptotic behaviour of all integer moments of a number of particles at any point $y\in\mathbb{Z}^d$ and of the particle population on $\mathbb{Z}^d$ according to the ratio $d/\alpha$.