Bellman-Harris processes and their application to branching random walks

In the talk, we consider critical catalytic branching random walk on an integer lattice having an arbitrary starting point. The main attention is paid to finding the asymptotic behavior of the probability of the particles presence at an arbitrary fixed point of the lattice as time tends to infinity. Moreover, the Yaglom type conditional limit theorems are established for the number of particles at each point of the lattice. The method of proof is based on introduction of an auxiliary Bellman-Harris branching process with particles of six types. An essential role is played by the proposed notion of "a hitting time with taboo" for a (non-branching) random walk on the lattice.
[1] Bulinskaya E.Vl. Limit Theorems for Local Particles Numbers in Branching Random Walk. Doklady RAN (in print).
[2] Bulinskaya E.Vl. The Hitting Times with Taboo for a Random Walk on an Integer Lattice. Prepublication de LPMA UPMC no.1456 (2011), 25 pp.; arXiv:1107.1074v1 [math.PR].