A Branching Random Walk with a Barrier

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.(.log(.e.tdF(t))). We show that the situation is black or white: 1. If b<eh,thenP(family line dies)=1. 2. If b>eh,thenP(family survives)>0. Define f(x):=E(number of members in the family.initial strengthx). We show that if b<eh, then there exists a positive constant C such that limx..e..xf(x)=C, where . is the smaller of the (at most) two positive roots of b.estdF(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.