The uniqueness of hierarchically extended backward solutions of the Wright-Fisher model

TL;DR

This paper introduces a novel regularisation scheme for backward solutions of the Wright-Fisher model's Kolmogorov equations, addressing boundary degeneracies and establishing their uniqueness in a biological context.

Contribution

It develops a regularising blow-up scheme for extended solutions of the backward Kolmogorov equation, proving their uniqueness despite boundary degeneracies.

Findings

Established a regularising transformation for solutions

Proved uniqueness of extended solutions

Addressed boundary degeneracy issues in the model

Abstract

The diffusion approximation of the Wright-Fisher model of population genetics leads to partial differentiable equations, the so-called Kolmogorov equations, with an operator that degenerates at the boundary. Standard tools do not apply, and in fact, solutions lack regularity properties. In this paper, we develop a regularising blow-up scheme for a certain class of solutions of the backward Kolmogorov equation, the iteratively extended global solutions presented in \cite{THJ5}, and establish their uniqueness. As the model describes the random genetic drift of several alleles at the same locus from a backward perspective, the singularities result from the loss of an allele. While in an analytical approach, this causes substantial difficulties, from a biological or geometric perspective, this is a natural process that can be analyzed in detail. The presented scheme regularises the solution via a tailored successive transformation of the domain.