无限测度$\mathbb{R}$-树上质量擦除 I：构造与极限定理
Mass erasure on infinitely measured $\mathbb{R}$-trees I: construction and limit theorems
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中文总结 AI 辅助
本文推广质量擦除理论至无限测度$\mathbb{R}$-树，构造离散无限测度上的半群算子，并证明Gromov-模糊收敛与质量擦除模糊收敛的等价条件，为超临界Galton-Watson和Lévy森林的不变性原理奠定基础。
中文摘要 AI 辅助
本文是两部分系列中的第一部分，旨在i)将Duquesne和Winkel发展的质量擦除理论推广到无限测度的设定，ii)应用该理论在最一般的设定下（树可能是超临界的且不满足Grey条件）获得Galton-Watson和Lévy森林的不变性原理。我们考虑配备有界有限测度的适当子类的$\mathbb{R}$-树，该子类称为离散无限测度。设$(T, d, \rho)$为完备可分$\mathbb{R}$-树，$\mu$为$T$上的离散无限测度，且$h > 0$。$h$-质量擦除子树通过移除所有$\mu$-质量严格小于$h$的边缘子树获得，我们证明所有$h$-质量擦除子树具有离散分支结构。质量擦除子树可配备相关测度，这产生算子族$(\mathscr{E}_h)_{h \geq 0}$，该算子族在离散无限测度空间上构成半群。这些算子可提升到适当的Gromov-模糊等距类空间。给定一列合适的等距类$\boldsymbol{\mu}_n = [T_n, d_n, \rho_n, \mu_n]$，$n \in \mathbb{N}$，若其质量擦除$(\mathscr{E}_h \boldsymbol{\mu}_n)_{n \in \mathbb{N}}$对所有$h > 0$在Gromov-模糊拓扑下收敛，则称它们在质量擦除意义下模糊收敛。我们证明通过对$(\boldsymbol{\mu}_n)_{n \in \mathbb{N}}$施加适当的紧性条件，Gromov-模糊收敛蕴含质量擦除意义下的模糊收敛。在适当的紧性下，我们识别出联系两种收敛概念的条件，并证明任一模式的收敛都蕴含质量擦除子树的局部Gromov-Hausdorff收敛。对Galton-Watson和Lévy森林的应用在第二部分中考虑。
英文摘要
This paper is the first in a two-part series, seeking to i) extend the theory of mass erasure developed by Duquesne and Winkel to a setting with infinite measures, and ii) apply this theory to obtain invariance principles for Galton--Watson and Lévy forests in the most general setting where the trees may be supercritical and not satisfy Grey's condition. We consider $\mathbb{R}$-trees equipped with a suitable subclass of boundedly finite measures, which we call discretely infinite measures. Let $(T, d, ρ)$ be a complete and separable $\mathbb{R}$-tree, let $μ$ be a discretely infinite measure on $T$, and let $h > 0$. The $h$-mass-erased subtree is obtained by removing all fringe subtrees with $μ$-mass strictly less than $h$, and we prove that all $h$-mass-erased subtrees have a discrete branching structure. The mass-erased subtrees may be equipped with associated measures, which gives rise to a family of operators $(\mathscr{E}_h)_{h \geq 0}$ which forms a semigroup on the space of discretely infinite measures. The operators may be lifted to an appropriate space of Gromov-vague isometry classes. Given a sequence $\boldsymbolμ_n = [T_n, d_n, ρ_n, μ_n]$, $n \in \mathbb{N}$, of suitable isometry classes, we say that they converge vaguely in the sense of mass erasure if their mass erasures $(\mathscr{E}_h \boldsymbolμ_n)_{n \in \mathbb{N}}$ converge in the Gromov-vague topology for all $h > 0$. We prove that by imposing suitable tightness conditions on $(\boldsymbolμ_n)_{n \in \mathbb{N}}$, Gromov-vague convergence implies vague convergence in the sense of mass erasure. Under appropriate tightness, we identify conditions which relate the two notions of convergence, and prove that either mode of convergence implies local Gromov--Hausdorff convergence of the mass-erased subtrees. The applications to Galton--Watson and Lévy forests are considered in part two.
发表机构
- Mathematical Institute & Department of Statistics, University of Oxford（牛津大学数学研究所与统计系）
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