A continuous time branching random walk on the lattice Z is considered in which individuals may produce children at the origin only. Assuming that the underlying random walk is symmetric and the offspring reproduction law is critical we prove a conditional limit theorem for the number of individuals at the origin.
Philippe Carmona;Yueyun Hu, 2014, The spread of a catalytic branching random walk, Annales de l Institut Henri Poincaré Probabilités et Statistiques, 50, 2, 10.1214/12-aihp529, https://doi.org/10.1214/12-aihp529.