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arXiv · 1407.3067

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

Julian Hofrichter·Tat Dat Tran·Jürgen Jost
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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 \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.

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Julian Hofrichter, Tat Dat Tran, Jürgen Jost. 2014-07-11. The uniqueness of hierarchically extended backward solutions of the Wright-Fisher model. https://doi.org/10.1080/03605302.2015.1116558

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