Consider a diploid population of N individuals undergoing random mating and mutation as in the infinite-allele Wright model. Choose a particular gene in generation Q for large Q, and let ft (0 ≦ t ≦ Q) be the frequency of the allelic type of the predecessor of that particular gene (or the gene itself) in generation t. By considering a ‘diffusion approximation' {xt} of {ft}, we find the distribution of the age of an allelic type now known to have frequency p, and of its distribution of frequencies since the allele came into existence. A novelty here is that the process {xt} is not a diffusion, but a process with jumps; it has x = 0 as an inaccessible entrance boundary but periodically jumps to it from the interior of [0, 1]. The formulas obtained are the same as those derived by Maruyama and Kimura, who used a totally different approach.

Let I(x, u) be the probability that two genes found a vector distance x apart are the same type in an infinite-allele selectively-neutral migration model with mutation rate u. The creatures involved inhabit an infinite of colonies, are diploid and are held at N per colony. Set in one dimension and in higher dimensions, where σ2 is the covariance matrix of the migration law (which is assumed to have finite fifth moments). Then in one dimension, in two dimensions, and in three dimensions uniformly for Here C0 is a constant depending on the migration law, K0(y) is the Bessel function of the second kind of order zero, and are the eigenvalues of σ2. For symmetric nearest-neighbor migrations, in one dimension and log mi in two. For is known in one dimension and C0 does not appear. In two dimensions, These results extend and make more precise earlier work of Malécot, Weiss and Kimura and Nagylaki.

An expression is found for the stationary density of the allele frequencies, in the infinitely-many alleles model. It is assumed that all alleles are neutral, that there is a constant mutation rate, and that the population is sufficiently large for the diffusion model to apply. Bounds on some moments are calculated from the density, and some applications are made to the problem of which allele is oldest in a population. A postulate made by Ewens (1972), concerning the distribution of allele numbers in a finite random sample from the neutral diffusion population, is shown to be correct.