On the sojourn time distribution of a random walk at a multidimensional lattice point

We consider critical symmetric branching random walks on a multidimensional lattice with continuous time and with the source of particle birth and death at the origin. We prove limit theorems on the distribution of the sojourn time of the underlying random walk at a point depending on the lattice dimension under the assumption of finite variance and under a condition leading to infinite variance of jumps. We study the limit distribution of the population of particles at the source for recurrent critical branching random walks.