Structure of the Particle Population for a Branching Random Walk with a Critical Reproduction Law |
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Authors: | Balashova Daria Molchanov Stanislav Yarovaya Elena |
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Institution: | 1.Department of Probability Theory, Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, Moscow, Russia ;2.Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC, USA ;3.Laboratory of Stochastic Analysis and its Applications, National Research University Higher School of Economics, Moscow, Russia ; |
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Abstract: | We consider a continuous-time symmetric branching random walk on the d-dimensional lattice, d ≥?1, and assume that at the initial moment there is one particle at every lattice point. Moreover, we assume that the underlying random walk has a finite variance of jumps and the reproduction law is described by a continuous-time Markov branching process (a continuous-time analog of a Bienamye-Galton-Watson process) at every lattice point. We study the structure of the particle subpopulation generated by the initial particle situated at a lattice point x. We replay why vanishing of the majority of subpopulations does not affect the convergence to the steady state and leads to clusterization for lattice dimensions d =?1 and d =?2. |
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