非均匀分支布朗运动的谱分析与k-脊分解：完全推动前沿中的谱系
Spectral analysis and $k$-spine decomposition of inhomogeneous branching Brownian motions. Genealogies in fully pushed fronts
AI总结：
本文研究完全推动型波动前沿中非均匀分支布朗运动的谱系结构，通过建立Yaglom律、脊分解和谱分析方法，证明了粒子谱系收敛于Brownian Coalescent Point Process，并确定了脊的不变测度与混合时间。
摘要：
我们考虑一个粒子系统，该系统执行一维二元分支布朗运动，具有空间依赖的分支速率、负漂移 $-μ$，并在到达 $0$ 时被杀死。更准确地说，粒子以速率 $r(x)=(1+W(x))/2$ 分支，其中 $W$ 是一个紧支撑的非负光滑函数，漂移 $μ$ 的选择使得系统在某种意义上是临界的。该粒子系统可被视为一个解析上可处理的波动前沿模型，用于描述合作种群入侵栖息地的内部机制。Birzu、Hallatschek 和 Korolev 最近的研究表明存在三类波动前沿：拉动型、半推动型和完全推动型前沿。这里我们专注于完全推动型区域。我们为该分支过程建立了 Yaglom 律，并利用矩方法证明了粒子的谱系收敛于 Brownian Coalescent Point Process。在实践中，BBM 的谱系被视为一个随机标记的度量测度空间，我们使用脊分解证明了其在 Gromov-弱拓扑下的收敛性。我们还对与 BBM 相关的微分算子进行了谱分解，以确定脊的不变测度及其混合时间。
英文摘要：
We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with space-dependent branching rate, negative drift $-μ$ and killed upon reaching $0$. More precisely, the particles branch at rate $r(x)=(1+W(x))/2,$ where $W$ is a compactly supported and non-negative smooth function and the drift $μ$ is chosen in such a way that the system is critical in some sense. This particle system can be seen as an analytically tractable model for fluctuating fronts, describing the internal mechanisms driving the invasion of a habitat by a cooperating population. Recent studies from Birzu, Hallatschek and Korolev suggest the existence of three classes of fluctuating fronts: pulled, semi pushed and fully pushed fronts. Here, we focus on the fully pushed regime. We establish a Yaglom law for this branching process and prove that the genealogy of the particles converges to a Brownian Coalescent Point Process using a method of moments. In practice, the genealogy of the BBM is seen as a random marked metric measure space and we use spinal decomposition to prove its convergence in the Gromov-weak topology. We also carry the spectral decomposition of a differential operator related to the BBM to determine the invariant measure of the spine as well as its mixing time.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
