Splitting trees with neutral Poissonian mutations I: Small families

We consider a supercritical branching population, where individuals have i.i.d. lifetime du-rations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate. We assume that individuals independently experience neutral mutations, at constant rate θ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously con-fers a brand new type, called allele or haplotype, to its carrier. The type carried by a mother at the time when she gives birth is transmitted to the newborn. We are interested in the sizes and ages at time t of the clonal families carrying the most abundant alleles or the oldest ones, as t→∞, on the survival event. Intuitively, the results must depend on how the mutation rate θ and the Malthusian parameter α> 0 compare. Hereafter, N ≡ Nt is the population size at time t, constants a, c are scaling constants, whereas k, k ′ are explicit positive constants which depend on the parameters of the model. When α> θ, the most abundant families are also the oldest ones, they have size cN1−θ/α and age t − a. When α < θ, the oldest families have age (α/θ)t+a and tight sizes; the most abundant familie