Individuals at the origin in the critical catalytic branching random walk

A continuous time branching random walk on the lattice 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. Keywords: catalytic branching random walk; critical two-dimensional Bellman-Harris process 1 Statement of problem and main results We consider the following modification of a standard branching random walk on ¡. Consider a population of individuals evolving as follows. The population is initiated at time t ¢ 0 by a single particle. Being outside the origin the particle performs a continuous time random walk on ¡ with infinitesimal transition matrix A ¢¤ £ a ¥ x ¦ y§¨ £ x © y�� � ¦ a ¥ 0 ¦ 0§� � 0¦ until the moment when it hits the origin. At the origin it spends an exponentially distributed time with parameter 1 and then either jumps to � ¢ a point y �� ¥ 0 � α § a ¥ 0 ¦ y § a � with probability 1 1 or dies with ¥ 0 ¦ 0§ probability α producing just before the death a random number of children ξ in accordance with offspring generating function de f ¢� � f s ¥ s § ξ ∞ ∑ fks ¢ k � 0 k� At the birth moment the newborn particles are located at the origin but after this moment behave independently and stochastically the same as the parent individual. The model we consider here is a particular case of the so-called branching random walk in catalytic medium. The longtime behavior of branching random walks in catalytic media of various types were investigated by a number of authors (see cf. (GKW99) and (W91) and the relevant references therein)