Wright-Fisher模型分层延拓向后解的唯一性
The uniqueness of hierarchically extended backward solutions of the Wright-Fisher model
摘要：
群体遗传学的赖特-费舍尔模型的扩散近似导致偏微分方程，即所谓的柯尔莫哥洛夫方程，其算子在边界处退化。标准工具不适用，事实上，解决方案缺乏规律性。本文对一类后向Kolmogorov方程的解，即\cite{THJ5}中的迭代扩张整体解，建立了一个正则化的爆破格式，并证明了其唯一性.由于该模型从向后的角度描述了同一基因座上几个等位基因的随机遗传漂变，因此奇异性是由等位基因的丢失引起的。虽然在分析方法中，这会造成很大的困难，但从生物学或几何学的角度来看，这是一个可以详细分析的自然过程。所提出的计划regurises通过定制的连续变换域的解决方案。
英文摘要：
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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