Operators Satisfying the Schur Condition and their Applications to the Branching Random Walks

Abstract

We consider continuous-time branching random walks on multidimensional lattices with birth and death of particles at the origin, the source of branching. The operators arising in the description of evolution of the mean numbers of particles in models of branching random walks are shown to satisfy the Schur condition. We prove theorems on the properties of the solutions of the differential equations with arbitrary difference operators satisfying the Schur condition. These results are applied to the study of solutions of the differential equations with the evolutionary operators arising in the models of branching random walks.