Recurrence and transience of multitype branching random walks

We study a discrete time Markov process with particles being able to perform discrete time random walks and create new particles, known as branching random walk (BRW). We suppose that there are particles of different types, and the transition probabilities, as well as offspring distribution, depend on the type and the position of the particle. Criteria of (strong) recurrence and transience are presented, and some applications (spatially homogeneous case, Lamperti BRW, many-dimensional BRW) are studied.