Branching Walks with a Finite Set of Branching Sources and Pseudo-sources

Abstract

AbstractBranching random walks play a key role in modeling the evolutionary processes with birth and death of particles depending on the structure of a medium. The branching random walk on a multidimensional lattice with a finite number of branching sources of three types is investigated. It is assumed that the intensities of branching in the sources can be arbitrary. The principal attention is paid to the analysis of spectral characteristics of the operator describing evolution of the mean numbers of particles both at an arbitrary point and on the entire lattice. The obtained results provide an explicit conditions for the exponential growth of the numbers of particles without any assumptions on jumps variance of the underlying random walk.KeywordsBranching random walksEquations in Banach spacesNon-homogeneous environmentsPositive eigenvaluesPopulation dynamics