YAGLOM LIMIT FOR CRITICAL NONLOCAL BRANCHING MARKOV PROCESSES

• 成果类型: Article
• 署名作者: Harris, Simon C.; Horton, Emma; Kyprtanou, Andreas E.; Wang, Minmin
• 署名单位: University of Auckland; Inria; University of Bath; University of Sussex
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1585
• 发表日期: 2022
• 页码: 2373-2408
• 关键词: stochastic methods; large numbers; asymptotic-behavior; general set; LAW; superprocesses; THEOREM

摘要

We consider the classical Yaglom limit theorem for a branching Markov process X = (X-t, t >= 0), with nonlocal branching mechanism in the setting that the mean semigroup is critical, that is, its leading eigenvalue is zero. In particular, we show that there exists a constant c(f ) such that Law(< f, X-t >/t < f, X-t > > 0) -> e(c)(f), t -> infinity, where e(c)(f ) is an exponential random variable with rate c(f ) and the con-vergence is in distribution. As part of the proof, we also show that the prob-ability of survival decays inversely proportionally to time. Although Yaglom limit theorems have recently been handled in the setting of branching Brow-nian motion in a bounded domain and superprocesses, (Probab. Theory Re-lated Fields 173 (2019) 999-1062; Electron. Commun. Probab. 23 (2018) 42), these results do not allow for nonlocal branching which complicates the analysis. Our approach and the main novelty of this work is based around a precise result for the scaled asymptotics for the kth martingale moments of X (rather than the Yaglom limit itself). We then illustrate our results in the setting of neutron transport for which the nonlocality is essential, comple-menting recent developments in this domain (Ann. Appl. Probab. 30 (2020) 2573-2612; Ann. Appl. Probab. 30 (2020) 2815-2845; SIAM J. Appl. Math. 81 (2021) 982-1001; Cox et al. (2021); J. Stat. Phys. 176 (2019) 425-455).