A transition density expansion for a multi-allele diffusion model

Abstract

An expansion in orthogonal polynomials is found for the transition density in a neutral multi-allele diffusion model where the mutation rates of allele typesA_i→A_jare assumed to beu_j(≥ O). The density is found when the mutation rate is positive for all allele types, and when some or all have zero mutation. The asymptotic conditional density is found for a mixture of positive and zero mutation rates.The infinite alleles limit with equal mutation is studied. Eigenfunctions of the process are derived and the frequency spectrum found. An important result is that the first eigenfunction depends only on the homozygosity.A density for the time to fixation with zero mutation is found for theKallele, and infinite alleles model.