论文标题
聚合和抽样分布的菌落扩散分布
Coalescence and sampling distributions for Feller diffusions
论文作者
论文摘要
考虑由正方程$ u_t（t，x）= \ tfrac {1} {2} {2} \ {x u（t，x）\} _ {xx} - α\ {x u（x u（t，t，x） $ u（0，x）=δ（x -x_0）$。 Feller引入并解决了该方程，以模拟独立繁殖个体的人群的增长。我们探索与Feller解决方案有关的重要合并过程。对于任何$α$和$ x_0> 0 $，我们计算了随机变量$ a_n（s; t）$的分布，该分布在过去的一个$ n $的样本中定义为有限的祖先数量$ s $，自从其起步以来，它一次是从一个feller扩散的无限范围人群中获取的。在亚临界扩散中，我们发现从时间$ t $ back的种群和样本合并树的分布，以非膨胀为$ t \ to \ infty $。在超临界扩散中，我们构建了一个有一个创始人并得出结合时间的分布的共聚树。
Consider the diffusion process defined by the forward equation $u_t(t, x) = \tfrac{1}{2}\{x u(t, x)\}_{xx} - α\{x u(t, x)\}_{x}$ for $t, x \ge 0$ and $-\infty < α< \infty$, with an initial condition $u(0, x) = δ(x - x_0)$. This equation was introduced and solved by Feller to model the growth of a population of independently reproducing individuals. We explore important coalescent processes related to Feller's solution. For any $α$ and $x_0 > 0$ we calculate the distribution of the random variable $A_n(s; t)$, defined as the finite number of ancestors at a time $s$ in the past of a sample of size $n$ taken from the infinite population of a Feller diffusion at a time $t$ since since its initiation. In a subcritical diffusion we find the distribution of population and sample coalescent trees from time $t$ back, conditional on non-extinction as $t \to \infty$. In a supercritical diffusion we construct a coalescent tree which has a single founder and derive the distribution of coalescent times.
