发表机构
University of Auckland; King’s College London; Centro de Investigación en Matemáticas (CIMAT)（奥克兰大学; 伦敦国王学院; 数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对连续时间Galton-Watson树在有限时间爆炸的现象，定义了爆炸脊柱，证明脊柱外出生呈泊松描述，爆炸前人口近似伽马分布，并证明重标度过程收敛于带移民的次临界连续状态分支过程，其时间反转具有马尔可夫性，关联合并过程为随机时间变换的Beta-合并过程。
AI 中文摘要
我们研究连续时间Galton-Watson树，其子代生成函数形式为 \begin{align} \label{eq:gen} f(s) = s - (1-s)^\alpha L(1-s), \end{align} 其中 $\alpha \in (0,1)$，且 $L:[0,1] \to [0,\infty)$ 在零处缓慢变化。具有该子代生成函数的随机过程在有限时间内爆炸。我们观察到，在时间 $T$ 爆炸的条件下，在每个更早时刻 $t < T$，存在一个唯一的粒子 $\xi_t$ 在时刻 $t$ 存活，它是爆炸时刻所有但有限多个粒子的祖先。我们将 $(\xi_t)_{t \in [0,T)}$ 称为爆炸脊柱，并证明脊柱之外的出生允许泊松描述，其中随着 $t \uparrow T$，这些出生变得更加频繁且规模更大。我们对爆炸前的人口规模进行了仔细研究，特别表明在爆炸附近，重新标度的人口近似服从伽马分布。更一般地，在时间 $T$ 爆炸的条件下，定义随机过程 $Z^\varepsilon:= (Z_t^\varepsilon)_{t \in \mathbb{R}}$，其中 \begin{align*} Z_t^\varepsilon:= E(\varepsilon e^{-t})N_{T-\varepsilon e^{-t}}, \qquad E(t) = \mathbf{P}(N_t = \infty). \end{align*} 我们证明当 $\varepsilon \downarrow 0$ 时，$Z^\varepsilon$ 在有限维分布上收敛到一个平稳的连续状态分支过程，该过程具有移民，其分支机制是次临界的，其移民机制对应于脊柱事件。最后，我们证明该极限过程具有马尔可夫时间反转，并且与其关联的合并过程是一个随机时间变换的Beta$(2-\alpha,\alpha)$-合并过程。
英文摘要
We study continuous-time Galton--Watson trees whose offspring generating function takes the form \begin{align} \label{eq:gen} f(s) = s - (1-s)^αL(1-s), \end{align} where $α\in (0,1)$ and where $L:[0,1] \to [0,\infty)$ is slowly varying at zero. Processes with this offspring generating function explode in finite time. We observe that, conditional on explosion at time $T$, at each earlier time $t < T$ there is a unique particle $ξ_t$ alive at time $t$ who is an ancestor of all but finitely many particles at the explosion time. We call $(ξ_t)_{t \in [0,T)}$ the spine to explosion, and show that the births off the spine admit a Poissonian description, where they become both more frequent and larger as $t \uparrow T$. We undertake a careful study of the size of the population leading up to explosion, showing in particular that near explosion, the rescaled population is approximately gamma distributed. More generally, conditional on explosion at time $T$, define a stochastic process $Z^\varepsilon := (Z_t^\varepsilon)_{t \in \mathbb{R}}$ by setting \begin{align*} Z_t^\varepsilon := E(\varepsilon e^{-t})N_{T-\varepsilon e^{-t}}, \qquad E(t) = \mathbf{P}(N_t = \infty). \end{align*} We show that as $\varepsilon \downarrow 0$, $Z^\varepsilon$ converges in finite-dimensional distributions to a stationary continuous-state branching process with immigration whose branching mechanism is subcritical and whose immigration mechanism corresponds to spine events. Finally, we show that this limiting process has a Markovian time reversal, and that the coalescent process associated with it is a stochastically time-changed Beta$(2-α,α)$-coalescent.
Comments37 pages, 4 figures
