Survival and complete convergence for a branching annihilating random walk

Birkner, Matthias and Callegaro, Alice and Černý, Jiří and Gantert, Nina and Oswald, Pascal. (2023) Survival and complete convergence for a branching annihilating random walk. Preprints Fachbereich Mathematik, 2023 (10).


Official URL: https://edoc.unibas.ch/95047/

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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 non-trivial ergodic equilibrium for $R$ sufficiently large. We also prove complete convergence for that case.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Wahrscheinlichkeitstheorie (Cerny)
12 Special Collections > Preprints Fachbereich Mathematik
UniBasel Contributors:Černý, Jiří and Oswald, Pascal
Item Type:Preprint
Publisher:Universität Basel
edoc DOI:
Last Modified:23 Jun 2023 13:03
Deposited On:23 Jun 2023 13:03

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