Branching Random Walks on $$\mathbb {Z}$$ Z with One Particle Generation Center and Symmetrically Located Absorbing Sources

Elena Filichkina () and Elena Yarovaya ()
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Elena Filichkina: Lomonosov Moscow State University
Elena Yarovaya: Lomonosov Moscow State University

Methodology and Computing in Applied Probability, 2024, vol. 26, issue 3, 1-16

Abstract: Abstract We consider a continuous-time branching random walk (BRW) on a one-dimensional lattice on which there is one center (the lattice point) of particle generation, called branching source. The generation of particles in the branching source is described by a Markov branching process. Some number (finite or infinite, depending on the problem formulation) of absorbing sources is located symmetrically around the branching source. For such configurations of sources we obtain necessary and sufficient conditions for the existence of an isolated positive eigenvalue of the evolution operator. It is shown that, under some additional assumptions, the existence of such an eigenvalue leads to an exponential growth of the number of particles in each point of the lattice.

Keywords: Branching random walks; Absorbing sources; Exponential growth of particle numbers; Evolution operator; 60J27; 60J80; 05C81; 60J85 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s11009-024-10097-8

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