Abstract:
A model of a branching random walk on $\mathbb Z^d$ with a periodic set of sources of reproduction and death of particles is studied. For this model, an asymptotic description of the propagation of the population of particles with time is obtained for the first time. The intensities of the sources may be different. The branching regime is assumed to be supercritical, and the tails of the jump distribution of the walk are assumed to be “light.” The main theorem establishes the Hausdorff-metric convergence of the properly normalized random cloud of particles that exist in the branching random walk at time $t$ to the limit set, as $t$ tends to infinity. This convergence takes place for almost all elementary outcomes of the event meaning nondegeneracy of the population of particles under study. The limit set in $\mathbb R^d$, called the asymptotic shape of the population, is found in an explicit form.
Keywords:branching random walk on periodic graphs, supercritical regime, propagation of population, Cramér condition, asymptotic shape of a branching random walk.
Citation:
E. Vl. Bulinskaya, “Propagation of Branching Random Walk on Periodic Graphs”, Noncommutative Analysis and Quantum Information Theory, Collected papers. Dedicated to Academician Alexander Semenovich Holevo on the occasion of his 80th birthday, Trudy Mat. Inst. Steklova, 324, Steklov Math. Inst., Moscow, 2024, 73–82; Proc. Steklov Inst. Math., 324 (2024), 66–74
\Bibitem{Bul24}
\by E.~Vl.~Bulinskaya
\paper Propagation of Branching Random Walk on Periodic Graphs
\inbook Noncommutative Analysis and Quantum Information Theory
\bookinfo Collected papers. Dedicated to Academician Alexander Semenovich Holevo on the occasion of his 80th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2024
\vol 324
\pages 73--82
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4370}
\crossref{https://doi.org/10.4213/tm4370}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4767948}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2024
\vol 324
\pages 66--74
\crossref{https://doi.org/10.1134/S0081543824010073}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85198101943}