AI 中文总结
研究泊松高尔顿 - 沃森树，通过确定参数导数、建立微分积分方程等方法，在特定条件下关联多种过程并证明总后代可积性。应用于稀疏非齐次随机图，得出连通组件数量极限及明确极限公式，扩展了相关公式适用范围。
AI 中文摘要
我们研究标准博雷尔型空间上后代核乘以标量参数时的泊松高尔顿 - 沃森树。在有限树上，确定两个参数值之间的拉东 - 尼科迪姆导数，并表明投影到总后代测度后它仍可测。在后代强度的一致界下，求导得出投影律的精确微分和积分方程，无需不可约性、可逆性或正特征函数。在有额外上下界均不为零的正特征函数时，将这些方程与无限脊柱树、均匀修剪、杜布变换和奥尔德斯 - 皮特曼提升过程相关联。对于一致有界的后代核，还证明了在谱亚临界区域总后代的一致指数可积性。作为应用，在博洛巴斯、扬森和里奥丹的图形核假设下，连通组件数量\(K_n\)满足\(K_n/n \to {\mathbb E}_{\pi}[1/T_u]\)依概率和\(L^1\)收敛，其中\(T_u\)是相关分支过程的总后代，\(1/\infty = 0\)。若\(q_u(x)\)是其从类型\(x\)的灭绝概率，重新生根和灭绝对偶给出明确极限\(\int_S q_u(x)\,\pi(dx) - {u\over 2}\int_{S\times S}\kappa(x,y)q_u(x)q_u(y)\,\pi(dx)\pi(dy)\)。这将有限类型和紧连续公式扩展到完整的BJR图形核设置，允许可分的非紧类型空间和可能无界或可约的核。
英文摘要
We study Poisson Galton--Watson trees on a standard Borel type space when the offspring kernel is multiplied by a scalar parameter. On finite trees, we identify the Radon--Nikodym derivative between two parameter values and show that it remains measurable after projection to the total progeny measure. Under a uniform bound on the offspring intensities, differentiation yields exact differential and integral equations for the projected laws without irreducibility, reversibility, or a positive eigenfunction. With an additional positive eigenfunction bounded above and away from zero, we relate these equations to an infinite spinal tree, uniform pruning, the Doob transform, and the Aldous--Pitman ascension process. For a uniformly bounded offspring kernel, we also prove uniform exponential integrability of the total progeny throughout the spectrally subcritical regime. As an application, under the graphical-kernel assumptions of Bollobas, Janson and Riordan, the number $K_n$ of connected components satisfies $K_n/n \to {\mathbb E}_π[1/T_u]$ in probability and in $L^1$, where $T_u$ is the total progeny of the associated branching process and $1/\infty=0$. If $q_u(x)$ is its extinction probability from type $x$, re-rooting and extinction duality give the explicit limit $$ \int_S q_u(x)\,π(dx) - {u\over 2}\int_{S\times S}κ(x,y)q_u(x)q_u(y)\,π(dx)π(dy). $$ This extends the finite-type and compact-continuous formulas to the full BJR graphical-kernel setting, allowing separable noncompact type spaces and kernels that may be unbounded or reducible.
Comments23 pages
