A Branching Random Walk with a Barrier

Abstract

Suppose that a child is likely to be weaker than its parent and a child who is too weak will not reproduce. What is the condition for a family line to survive? Let $b$ denote the mean number of children a viable parent will have; we suppose that this is independent of strength as long as strength is positive. Let $F$ denote the distribution of the change in strength from parent to child, and define $h = \sup_\theta(-\log(\int e^{\theta t} dF(t)))$. We show that the situation is black or white: 1. If $b < e^h, \text{then} P(\text{family line dies}) = 1$. 2. If $b > e^h, \text{then} P(\text{family survives}) > 0$. Define $f(x) := E(\text{number of members in the family} \mid \text{initial strength} x)$. We show that if $b < e^h$, then there exists a positive constant $C$ such that $\lim_{x \rightarrow \infty}e^{- \alpha x}f(x) = C$, where $\alpha$ is the smaller of the (at most) two positive roots of $b \int e^{st} dF(t) = 1$. We also find an explicit expression for $f(x)$ when the walk is on a lattice and is skip-free to the left. This process arose in an analysis of rollback-based simulation, and these results are the foundation of that analysis.