求解选择–重组方程：选择与重组下的祖先谱系
Solving the selection-recombination equation: Ancestral lines under selection and recombination
AI总结：
针对任意数量中性位点连锁一个受选择位点的选择–重组方程，论文借助祖先选择–重组图构建递归积分与多重等价随机过程，证明其正向对偶并给出确定性解及 Markov 半群的闭式随机表示。
摘要：
确定性选择–重组方程描述了在大数定律机制下，种群遗传类型组成在选择与重组作用下的演化。迄今为止，其显式解似乎仍遥不可及；仅在三个位点且选择作用于其中一个位点的特殊情形下，人们找到了近似解，而且没有明显的推广路径。针对任意数量的中性位点与一个受选择位点相连锁的情形，我们同时采用解析方法和概率的、谱系学方法。这给出了解的递归积分表示。从祖先选择–重组图的一个变体出发，我们发展出一种高效的谱系结构；等价地，它可以表示为一个加权划分过程、一族带启动与重置的 Yule 过程，以及一族启动过程。我们证明它们与时间正向微分方程的解对偶，从而获得确定性解的随机表示，并得到闭式的 Markov 半群。
英文摘要：
The deterministic selection-recombination equation describes the evolution of the genetic type composition of a population under selection and recombination in a law of large numbers regime. So far, an explicit solution has seemed out of reach; only in the special case of three sites with selection acting on one of them has an approximate solution been found, but without an obvious path to generalisation. We use both an analytical and a probabilistic, genealogical approach for the case of an \emph{arbitrary} number of neutral sites linked to one selected site. This leads to a recursive integral representation of the solution. Starting from a variant of the ancestral selection-recombination graph, we develop an efficient genealogical structure, which may, equivalently, be represented as a weighted partitioning process, a family of Yule processes with initiation and resetting, and a family of initiation processes. We prove them to be dual to the solution of the differential equation forward in time and thus obtain a stochastic representation of the deterministic solution, along with the Markov semigroup in closed form.
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