The Infinitely-Many-Sites Model as a Measure-Valued Diffusion

The infinitely-many-sites model (with no recombination) is reformulated, with sites labelled by elements of [0, 1] and "type" space E = 0, 1Z₊. A gene is of type x = (x₀, x₁,…) ∈ E if x₀, x₁, … is the sequence of sites at which mutations have occurred in the line of descent of that gene. The model is approximated by a diffusion process taking values in P⁰ₐ(E), the set of purely atomic Borel probability measures μ on E with the property that the locations of every n ≥ 1 atoms of μ form a family tree, and the diffusion is shown to have a unique stationary distribution μ̃. The principal object of investigation is the μ̃(dμ)-expectation of the probability that a random sample from a population with types distributed according to μ has a given tree structure. Ewens' (1972) sampling formula and Watterson's (1975) segregating-sites distribution are obtained as corollaries.