SPECTRAL ANALYSIS AND K-SPINE DECOMPOSITION OF INHOMOGENEOUS BRANCHING BROWNIAN MOTIONS. GENEALOGIES IN FULLY PUSHED FRONTS

• 成果类型: Article
• 署名作者: Schertzer, Emmanuel; Tourniaire, Julie
• 署名单位: University of Vienna; Universite Marie et Louis Pasteur; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1736
• 发表日期: 2025
• 页码: 1382-1433
• 关键词: stochastic methods; coalescent

摘要

We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with space-dependent branching rate, negative drift-mu 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 nonnegative smooth function and the drift mu 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, semipushed 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 out 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.