Periodic Multitype Symmetric Branching Random Walks on Z^d

We consider the model of symmetric branching continuous-time random walks on the lattice Zᵈ with n types of particles and periodically located branching sources. It is assumed that initially there is only one particle of type Tₛ at some point. For this process, we construct a periodic operator describing the evolution of the mean number of particles of type Tⱼ and study its spectral properties. We also obtain the asymptotics of the mean number of particles of type Tⱼ at a fixed point of the lattice as t →∞.