AI 中文总结
研究临界分支随机游走及其蛇的标度极限，在离散Sheu假设下证明重标度离散蛇收敛到ψ-布朗蛇，并应用于值域极限到反射ψ-布朗仙人掌。
AI 中文摘要
我们研究取值于 $\mathbb R^d$ 的分支随机游走（简称BRWs）的标度极限，设定如下：索引树为具有临界后代分布的Galton-Watson树，假设其在适当重标度下收敛到临界 $\psi$-Lévy 树；在给定索引树的条件下，BRW的跳跃在同一兄弟组内可能具有依赖性，但不同兄弟组之间相互独立，且其分布是中心化的但可能变化（例如可能依赖于整棵索引树），然而游走的典型值仍保持在某个高斯分布混合的吸引域内。在这些假设以及一个必要的附加假设（我们称之为离散Sheu假设）下，我们证明重标度的离散蛇在函数意义上收敛到由Le Gall & Le Jan (1998) 和 D.~& Le Gall (2002) 引入的 $\psi$-布朗蛇。该方法使用了具有独立意义的BRW耦合结果。作为应用，我们给出了取值于 $\mathtt b$-叉树的BRW的值域对标度极限，收敛到反射 $\psi$-布朗仙人掌，后者是Curien, Le Gall & Miermont (2013) 引入的布朗仙人掌的一个变体。
英文摘要
We study scaling limits of $\mathbb R^d$-valued branching random walks (BRWs for short) in the following setting: indexing-trees are Galton-Watson trees with critical offspring distribution which are assumed, when appropriately rescaled, to converge to a critical $ψ$-Lévy tree; conditional on the indexing tree, the jumps of the BRW may exhibit dependence within the same sibling group, but distinct sibling groups are independent, and their distributions are centered but may vary (and for instance depend on the whole indexing tree), with a typical value of the walk nevertheless remaining within the domain of attraction of a mixture of Gaussian distributions. Under these assumptions and a necessary additional assumption, which we call the discrete Sheu assumption, we show that rescaled discrete snakes converge functionally to the $ψ$-Brownian snake introduced by Le Gall & Le Jan (1998) and D.~& Le Gall (2002). The method uses a coupling result for BRWs of independent interest. An application is given concerning the scaling limits of the range of BRWs that take their values in the $\mathtt b$-ary tree to the reflected $ψ$-Brownian cactus which is a variant of the Brownian cactus introduced by Curien, Le Gall & Miermont (2013).
