One Limit Theorem for Branching Random Walks

Abstract

Theory of Probability &Its Applications, Volume 68, Issue 4, Page 630-642, February 2024.
The foundations of the general theory of Markov random processes were laid by A.N. Kolmogorov. Such processes include, in particular, branching random walks on lattices $\mathbf{Z}^d$, $d \in \mathbf{N}$. In the present paper, we consider a branching random walk where particles may die or produce descendants at any point of the lattice. Motion of each particle on $\mathbf{Z}^d$ is described by a symmetric homogeneous irreducible random walk. It is assumed that the branching rate of particles at $x \in \mathbf{Z}^d$ tends to zero as $\|x\| \to \infty$, and that an additional condition on the parameters of the branching random walk, which gives that the mean population size of particles at each point $\mathbf{Z}^d$ grows exponentially in time, is met. In this case, the walk generation operator in the right-hand side of the equation for the mean population size of particles undergoes a perturbation due to possible generation of particles at points $\mathbf{Z}^d$. Equations of this kind with perturbation of the diffusion operator in $\mathbf{R}^2$, which were considered by Kolmogorov, Petrovsky, and Piskunov in 1937, continue being studied using the theory of branching random walks on discrete structures. Under the above assumptions, we prove a limit theorem on mean-square convergence of the normalized number of particles at an arbitrary fixed point of the lattice as $t\to\infty$.