Immortal particle for a catalytic branching process

Abstract. We study the existence and some asymptotic properties of a conservative branching particle system for which birth and death are triggered by contact with a set. Sufficient conditions for the process to be non-explosive are given, solving a long standing open problem. With probability one, it is shown that only one ancestry survives. In special cases, the evolution of the surviving particle is studied and for a two particle system on a half line we derive explicitly the transition function of a chain representing the position at successive branching times. 1