空间分支过程向α-稳定连续状态分支过程的收敛：半推进前沿的谱系
Convergence of spatial branching processes to $α$-stable CSBPs: Genealogy of semi-pushed fronts
AI总结：
研究\(\mathbb{R}^d\)无界域上非齐次分支扩散，提出条件使规模过程和谱系收敛到\(\alpha\)-稳定连续状态分支过程。通过矩收敛条件及递归公式给出充分条件，应用于一维分支布朗运动，证明半推进 regime 中谱系收敛。
摘要：
我们考虑在\(\mathbb{R}^d\)的无界域上的非齐次分支扩散，并提出一个简单条件，在此条件下我们期望规模过程（即粒子数）和系统的谱系收敛到\(\alpha\)-稳定连续状态分支过程的相应部分，其中\(\alpha\in(1,2)\)。此条件可视为高尔顿 - 沃森过程后代分布尾部正则变化的经典假设的空间类似物。我们通过提供一组充分条件迈出了建立此结果的第一步，在这些条件下，被视为捕获种群位置和谱系结构的随机标记度量测度空间的分支扩散收敛到\(\alpha\)-稳定谱系。这些条件基于过程矩的收敛，可通过递归公式有效计算。我们将此框架应用于具有非齐次分支率和负漂移的一维分支布朗运动。该模型由图尔尼亚尔引入作为研究波动推进前沿内部动态的玩具模型。通过使用我们的一般条件集，我们证明了在半推进 regime 中过程谱系的收敛，这是比尔祖、哈拉切克和科罗廖夫所推测成立的。
英文摘要：
We consider an inhomogeneous branching diffusion on an unbounded domain of $\mathbb{R}^d$ and propose a simple condition under which we expect the size process (i.e., the number of particles) and the genealogy of the system to converge to those of an $α$-stable continuous-state branching process, with $α\in(1,2)$. This condition can be seen as the spatial analogue of the classical assumption that the tail of the offspring distribution of a Galton--Watson process is regularly varying. We make a first step towards establishing this result by providing a set of sufficient conditions under which the branching diffusion, seen as a random marked metric measure space that captures both the positions and the genealogical structure of the population, converges to an $α$-stable genealogy. These conditions are based on the convergence of the moments of the process, which can be efficiently computed via recursive formulas. We apply this framework to a one-dimensional branching Brownian motion with inhomogeneous branching rate and negative drift. This model was introduced by Tourniaire as a toy model to investigate the internal dynamics of fluctuating pushed fronts. By using our general set of conditions we prove convergence of the genealogy of the process in the semipushed regime, which was conjectured to hold by Birzu, Hallatschek, and Korolev.
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