On the Mean Number of Particles of a Branching Random Walk on ℤ d with Periodic Sources of Branching

We consider a continuous-time branching random walk on 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 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 → ∞.