On Random Walks with Geometric Lifetimes

We consider the following stochastic process. Initially, there is a single particle placed at the origin of Z. This particle moves as a discrete-time simple random walk on Z, which takes one step to the right with probability p. Furthermore, the particle has a probability of death (1−α) before each step. Whenever this particle returns to the origin, it gives birth to a new particle, which performs the same dynamics. For which values of the pair (p,α) does this model survive with positive probability? Can we obtain the survival probability as a function of (p,α)? By answering these questions, we review some fundamental concepts and tools in probability, such as the return times of a simple random walk on the integers, probability generating functions, branching processes, and coupling.