On the Mean Number of Particles of a Branching Random Walk on ${{\mathbb{Z}}^{d}}$ with Periodic Sources of Branching

Abstract—We consider a continuous-time branching random walk on ${{\mathbb{Z}}^{d}}$, where the particles are born and die on a periodic set of points (sources of branching). The spectral properties of the evolution operator for the mean number of particles at an arbitrary point of ${{\mathbb{Z}}^{d}}$ are studied. This operator is proved to have a positive spectrum, which leads to an exponential asymptotic behavior of the mean number of particles as $t \to \infty $.