Critical branching random walks on low-dimensional lattices

We consider branching random walks with continuous time on integer lattices such that the particles born and die at a unique point. Under the assumption that the walk is symmetric and homogeneous, we derive integral and differential equations for the dynamics of local probabilities of continuation of the process in arbitrary nodes of the lattice, as well as probabilities of survival of the population of particles, for lattices of any dimension. In the critical case, we study the asymptotic behaviour, as $t\to\infty$, of local probabilities, probabilities of survival of the population of particles, and conditional distributions of the population size on $\mathbf Z$ and $\mathbf Z^2$.