Julien Berestycki, Jiaqi Liu, Bastien Mallein, Jason Schweinsberg
{"title":"The Yaglom limit for branching Brownian motion with absorption and slightly subcritical drift","authors":"Julien Berestycki, Jiaqi Liu, Bastien Mallein, Jason Schweinsberg","doi":"arxiv-2409.08789","DOIUrl":null,"url":null,"abstract":"Consider branching Brownian motion with absorption in which particles move\nindependently as one-dimensional Brownian motions with drift $-\\rho$, each\nparticle splits into two particles at rate one, and particles are killed when\nthey reach the origin. Kesten (1978) showed that this process dies out with\nprobability one if and only if $\\rho \\geq \\sqrt{2}$. We show that in the\nsubcritical case when $\\rho > \\sqrt{2}$, the law of the process conditioned on\nsurvival until time $t$ converges as $t \\rightarrow \\infty$ to a\nquasi-stationary distribution, which we call the Yaglom limit. We give a\nconstruction of this quasi-stationary distribution. We also study the\nasymptotic behavior as $\\rho \\downarrow \\sqrt{2}$ of this quasi-stationary\ndistribution. We show that the logarithm of the number of particles and the\nlocation of the highest particle are of order $\\epsilon^{-1/3}$, and we obtain\na limit result for the empirical distribution of the particle locations.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"27 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.08789","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Consider branching Brownian motion with absorption in which particles move
independently as one-dimensional Brownian motions with drift $-\rho$, each
particle splits into two particles at rate one, and particles are killed when
they reach the origin. Kesten (1978) showed that this process dies out with
probability one if and only if $\rho \geq \sqrt{2}$. We show that in the
subcritical case when $\rho > \sqrt{2}$, the law of the process conditioned on
survival until time $t$ converges as $t \rightarrow \infty$ to a
quasi-stationary distribution, which we call the Yaglom limit. We give a
construction of this quasi-stationary distribution. We also study the
asymptotic behavior as $\rho \downarrow \sqrt{2}$ of this quasi-stationary
distribution. We show that the logarithm of the number of particles and the
location of the highest particle are of order $\epsilon^{-1/3}$, and we obtain
a limit result for the empirical distribution of the particle locations.