Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
64/2014

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

Julian Hofrichter, Tat Dat Tran and Jürgen Jost

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.

Received:
Jul 12, 2014
Published:
Jul 16, 2014

Related publications

inJournal
2016 Journal Open Access
Julian Hofrichter, Tat Dat Tran and Jürgen Jost

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

In: Communications in partial differential equations, 41 (2016) 3, pp. 447-483