The uniqueness of hierarchically extended backward solutions of the Wright–Fisher model. Issue 3 (3rd March 2016)

• Record Type: Journal Article
• Authors: Hofrichter, Julian; Tran, Tat Dat; Jost, Jürgen
• Is Part Of: Communications in partial differential equations. Volume 41:Issue 3(2016)
• Page Start: 447
• Page End: 483
• Publication Date: 2016-03-03
• Subjects: Backward Kolmogorov equation -- blow-up of solutions -- global solution -- loss of alleles -- random genetic drift -- Wright-Fisher model; 35K65 -- 35J70 -- 60J70; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2015.1116558
• Languages: English
• ISSNs: 0360-5302

Abstract

ABSTRACT: The diffusion approximation of the Wright-Fisher model of population genetics leads to partial differentiable equations, the Kolmogorov forward and backward equations, with a leading term that degenerates at the boundary. This degeneracy has the consequence that standard PDE tools do not apply, and solutions lack regularity properties. In this paper, we develop a regularizing blow-up scheme for the iteratively extended global solutions of the backward Kolmogorov equation presented in a previous paper, which are constructed from a known class of solutions, and establish their uniqueness for the stationary case. As the model describes the random genetic drift of several alleles at the same locus from a backward perspective, the occurring singularities result from the loss of an allele. While in an analytical approach, this provides substantial difficulties, from a biological or geometric perspective, this is a natural process that can be analyzed in detail. The presented scheme regularizes the solution via a carefully constructed iterative transformation of the domain.