Structure of the particle population for a branching random walk with a\n critical reproduction law

2019/03/06 by Daria Balashova, Balashova, Daria, Stanislav Molchanov +3
Mathematics · Physics and Astronomy · #60F10 #60J35 #60J80 #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1903.02284

openalex publication_date 2019/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a continuous-time symmetric branching random walk on the\nd-dimensional lattice, d\≥ 1, and assume that at the initial moment there\nis one particle at every lattice point. Moreover, we assume that the underlying\nrandom walk has a finite variance of jumps and the reproduction law is\ndescribed by a critical Bienamye-Galton-Watson process at every lattice point.\nWe study the structure of the particle subpopulation generated by the initial\nparticle situated at a lattice point x. We answer why vanishing of the\nmajority of subpopulations does not affect the convergence to the steady state\nand leads to clusterization for lattice dimensions d=1 and d=2.\n

Related

• Heavy-Tailed Branching Random Walks on Multidimensional Lattices. A\n Moment Approach
• Heavy-tailed branching random walks on multidimensional lattices. A moment approach
• Survival Analysis of Particle Populations in Branching Random Walks
• Branching Random Walks in a Random Killing Environment with a Single Reproduction Source
• Branching Random Walks with One Particle Generation Center and Possible Absorption at Every Point