AI 中文总结
该研究定义了带测度$\bb{R}$-树的质量擦除算子与对应收敛性，证明其弱于Gromov-弱收敛并构造完备度量空间，还刻画了经质量擦除保持不变的（次）临界GW森林，建立其向Lévy森林收敛的极限定理，拓展了已有研究范围。
AI 中文摘要
设$h>0$。对于一棵带根$\rho$、装备有限Borel测度$\nu$的完备可分$\bb{R}$-树$(T,d)$，我们通过移除$T$中所有质量小于$h$的边缘子树来定义$h$-质量擦除树，并为其配备合适的测度，使得擦除算子族$(\bb{E}_h)_{h\b0}$构成一个关于Gromov-弱拓扑连续的半群。\n 随后，我们称序列$\boldsymbol{\nu}_n=(T_n,d_n,\rho_n,\nu_n)$（$n\bn\bb{N}$）依质量擦除意义收敛，若对所有$h\b0$，$(\bb{E}_h\boldsymbol{\nu}_n)_{n\bn\bb{N}}$都依Gromov-弱拓扑收敛。该收敛概念严格弱于Gromov-弱收敛，我们建立了关联这两种收敛概念的判定准则，并定义了一个可度量化质量擦除意义下收敛性的距离函数。通过将带测度$\bb{R}$-树的概念扩展到允许边界（无穷测地线的远端）上承载质量，我们得到了一个完备度量空间。\n 接下来，我们将满足再生分支性质的有限型随机树（即带边长的离散树）识别为一类特殊的带测度（次）临界GW森林。随后证明这类树在质量擦除操作下保持不变，并显式计算了这些经质量擦除的GW森林的分布。最后，我们建立了这类带测度（次）临界GW森林收敛到标准带测度Lévy森林（即总质量与连续状态分支过程的总种群同分布的森林）的极限定理。该结果通过关键使用质量擦除意义下的收敛，涵盖了有界变差的情形，拓展了此前研究的范围。
英文摘要
Let $h>0$. For a complete and separable $\mathbb{R}$-tree $(T,d)$ equipped with a root $ρ$ and a finite Borel measure $μ$, we define the $h$-mass-erased tree by removing from $T$ all fringe subtrees of mass less than $h$ and we equip it with a suitable measure such that the erasure operators $(\mathcal{E}_h)_{h\ge 0}$ form a semigroup that is continuous for the Gromov-weak topology. Then, we say that a sequence $\boldsymbolμ_n=(T_n,d_n,ρ_n,μ_n)$, $n\in\mathbb{N}$, converges in the sense of mass erasure if $(\mathcal{E}_h\boldsymbolμ_n)_{n\in\mathbb{N}}$ converges Gromov-weakly for all $h\ge 0$. This notion of convergence is strictly weaker than Gromov-weak convergence and we establish criteria to relate the two notions. We define a distance function that metrizes convergence in the sense of mass erasure. By extending the notion of measured $\mathbb{R}$-trees to allow mass on the boundary (the far ends of infinite geodesics), we obtain a complete metric space. Next, we identify random trees of finite type (that is, discrete trees with edge lengths) satisfying the regenerative branching property as a specific class of measured (sub)critical GW forests. We then show that this class of trees is preserved by mass erasure and we compute the law of these mass-erased GW forests explicitly. Finally, we establish a limit theorem for these measured (sub)critical GW forests to converge to standard measured Lévy forests, i.e. those whose total mass has the same distribution as the total population of a continuous-state branching process. This includes cases with bounded variation by crucially using the convergence in the sense of mass erasure and it extends the cases studied previously.
Comments83 pages
