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

Julian Hofrichter, Tat Dat Tran, and Jürgen Jost

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Submission date: 12. Jul. 2014
Pages: 35
published in: Communications in partial differential equations, 41 (2016) 3, p. 447-483 
DOI number (of the published article): 10.1080/03605302.2015.1116558
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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 [?], 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.