Splitting trees with neutral Poissonian mutations I: Small families

We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate bb. Such a genealogical tree is usually called a splitting tree [9], and the population counting process (Nt;t≥0)(Nt;t≥0) is a homogeneous, binary Crump–Mode–Jagers process.We assume that individuals independently experience mutations at constant rate θθ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called an allele, to its carrier. We are interested in the allele frequency spectrum at time tt, i.e., the number A(t)A(t) of distinct alleles represented in the population at time tt, and more specifically, the numbers A(k,t)A(k,t) of alleles represented by kk individuals at time tt, k=1,2,…,Ntk=1,2,…,Nt.We mainly use two classes of tools: coalescent point processes, as defined in [15], and branching processes counted by random characteristics, as defined in [11] and [13]. We provide explicit formulae for the expectation of A(k,t)A(k,t) conditional on population size in a coalescent point process, which apply to the special case of splitting trees. We separately derive the a.s. limits of A(k,t)/NtA(k,t)/Nt and of A(t)/NtA(t)/Nt thanks to random characteristics, in the same vein as in [19].Last, we separately compute the expected homozygosity by applying a method introduced in [14], characterizing the dynamics of the tree distribution as the origination time of the tree moves back in time, in the spirit of backward Kolmogorov equations.