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JOURNALS // Teoriya Veroyatnostei i ee Primeneniya // Archive

Teor. Veroyatnost. i Primenen., 2023 Volume 68, Issue 4, Pages 779–795 (Mi tvp5609)

This article is cited in 6 papers

On one limit theorem for branching random walks

N. V. Smorodinaabc, E. B. Yarovayadc

a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
b Saint Petersburg State University
c Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
d Lomonosov Moscow State University

Abstract: 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 $t\to\infty$.

Keywords: branching random walk, the Kolmogorov equations, martingale, limit theorems.

Received: 17.05.2023
Accepted: 30.06.2023

DOI: 10.4213/tvp5609


 English version:
Theory of Probability and its Applications, 2024, 68:4, 630–642

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