A dual process for the coupled Wright-Fisher diffusion
Martina Favero, Henrik Hult, Timo Koski
TL;DR
This work derives a rigorous duality between the multi-locus, multi-allele coupled Wright-Fisher diffusion and an ancestral dual process built from L coupled ancestral selection graphs. Using a generator-based approach, the authors identify a jump-markov dual \\mathbf{N} on \\mathbb{N}^M with coalescence, mutation, single-branching, and novel double-branching events, whose rates are governed by a function k that encodes the stationary structure and selection across loci. In the special case of two loci and two alleles with parent-independent mutation, the dual rates and the diffusion's stationary density admit explicit forms, including representations via Beta and confluent hypergeometric functions. The framework extends to a general multi-locus setting and provides a path toward expansion of diffusion transition densities and inference schemes by leveraging dual genealogies. An explicit monotone coupling in the Appendix offers practical bounds on the dual process size, aiding numerical and analytical exploration of the model.
Abstract
The coupled Wright-Fisher diffusion is a multi-dimensional Wright-Fisher diffusion for multi-locus and multi-allelic genetic frequencies, expressed as the strong solution to a system of stochastic differential equations that are coupled in the drift, where the pairwise interaction among loci is modelled by an inter-locus selection. In this paper, an ancestral process, which is dual to the coupled Wright-Fisher diffusion, is derived. The dual process corresponds to the block counting process of coupled ancestral selection graphs, one for each locus. Jumps of the dual process arise from coalescence, mutation, single-branching, which occur at one locus at the time, and double-branching, which occur simultaneously at two loci. The coalescence and mutation rates have the typical structure of the transition rates of the Kingman coalescent process. The single-branching rate not only contains the one-locus selection parameters in a form that generalises the rates of an ancestral selection graph, but it also contains the two-locus selection parameters to include the effect of the pairwise interaction on the single loci. The double-branching rate reflects the particular structure of pairwise selection interactions of the coupled Wright-Fisher diffusion. Moreover, in the special case of two loci, two alleles, with selection and parent independent mutation, the stationary density for the coupled Wright-Fisher diffusion and the transition rates of the dual process are obtained in an explicit form.