发表机构
Australian National University（澳大利亚国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对小突变率下的费勒型中性多类型分支扩散，通过沿特征线求解拉普拉斯变换的前向柯尔莫哥洛夫方程，得到首阶坍缩到平稳左特征向量线密度及一阶矩近似，并给出临界时间与有效性条件。
AI 中文摘要
在突变率较小的极限下，找到了类似费勒中性的多类型分支扩散 $\big{(}\mathbf{X}(t)\big{)}_{t \in \mathbb{R}_{\ge 0}}$ 的近似解。所采用的方法涉及通过沿特征线积分来求解拉普拉斯变换后的前向柯尔莫哥洛夫方程的近似。在总突变率尺度 $\theta$ 的首阶，超临界扩散被发现坍缩到一条与速率矩阵的平稳左特征向量对齐的线密度上，该过程在临界时间 $t_{\rm c}$ 处发生快速行为变化，且 $t_{\rm c}$ 对 $\theta$ 有弱对数依赖性。还确定了 $\mathbf{X}(t)$ 的密度和矩的一阶 $\theta$ 近似。一阶近似对应于在样本的合并树中允许最多一次突变，在超临界情况下对 $t < t_{\rm c}$ 有效，并且更普遍地对临界和亚临界情况有效。
英文摘要
Approximate solutions are found for Feller-like neutral multi-type branching diffusions $\big{(}\mathbf{X}(t)\big{)}_{t \in \mathbb{R}_{\ge 0}}$ in the limit of small mutation rates. The method employed involves solving approximations to the Laplace transformed forward Kolmogorov equation by integrating along characteristics. To leading order in the scale $θ$ of the overall mutation rate the super-critical diffusion is found to collapse onto a line density aligned with the stationary left eigenvector of the rate matrix following a rapid change of behaviour at a critical time $t_{\rm c}$, which has a weak logarithmic dependence on $θ$. First order in $θ$ approximations to the density and moments of $\mathbf{X}(t)$ are also determined. The first-order approximation corresponds to allowing at most one mutation in the coalescent tree of a sample and is valid for $t < t_{\rm c}$ in the supercritical case, and more generally for the critical and sub-critical cases.
Comments35 pages, 1 figure
