SURVIVAL AND COMPLETE CONVERGENCE FOR A BRANCHING ANNIHILATING RANDOM WALK

• 成果类型: Article
• 署名作者: Birkner, Matthias; Callegaro, Alice; Cerny, Jiri; Gantert, Nina; Oswald, Pascal
• 署名单位: Johannes Gutenberg University of Mainz; Technical University of Munich; University of Basel
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2105
• 发表日期: 2024
• 页码: 5737-5768
• 关键词: interacting particle-systems; traveling-waves; probability; extinction

摘要

We study a discrete-time branching annihilating random walk (BARW) on the d-dimensional lattice. Each particle produces a Poissonian number of offspring with mean mu which independently move to a uniformly chosen site within a fixed distance R from their parent's position. Whenever a site is occupied by at least two particles, all the particles at that site are annihilated. We prove that for any mu > 1 the process survives when R is sufficiently large. For fixed R we show that the process dies out if mu is too small or too large. Furthermore, we exhibit an interval of mu -values for which the process survives and possesses a unique nontrivial ergodic equilibrium for R sufficiently large. We also prove complete convergence for that case.