A hierarchical extension scheme for backward solutions of the Wright–Fisher model
Julian Hofrichter, Tat Dat Tran,and Jürgen Jost
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Submission date: 19. Jun. 2014
Keywords and phrases: Wright-Fisher model, random genetic drift, backward Kolmogorov equation
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We develop an iterative global solution scheme for the backward Kolmogorov equation of the diffusion approximation of the Wright-Fisher model of population genetics. That model describes the random genetic drift of several alleles at the same locus in a population from a backward perspective. The key of our scheme is to connect the solutions before and after the loss of an allele. Whereas in an approach via stochastic processes or partial differential equations, such a loss of an allele leads to a boundary singularity, from a biological or geometric perspective, this is a natural process that can be analyzed in detail. A clarification of the role of the boundary resolves certain uniqueness issues and enlucidates the construction of hierarchical solutions.