Yaglom limit for critical nonlocal branching Markov processes
Yaglom limit for critical nonlocal branching Markov processes
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DOI：
10.1214/22-aop1585
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发表时间：
2022-11
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通讯作者：
S. Harris;E. Horton;A. Kyprianou;Minmin Wang
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作者：
S. Harris;E. Horton;A. Kyprianou;Minmin Wang
研究了分支马氏过程X =（Xt，t ≥ 0）的经典Yaglom极限定理，其中分支机制是非局部的，且平均半群是临界的，即其首特征值为零.特别地，我们证明了存在一个常数c（f），使得其中ec（f）是速率为c（f）的指数随机变量，并且收敛是分布的。作为证明的一部分，我们还表明，生存的概率与时间成反比衰减。虽然Yaglom极限定理最近在有界域和超过程中的分支布朗运动的背景下进行了处理，[35，36]，但这些结果不允许非局部分支，这使分析变得复杂。我们的方法和这项工作的主要新奇是基于一个精确的结果为缩放渐近的第k次鞅时刻的X（而不是Yaglom限制本身）。然后，我们在中子输运的设置中说明我们的结果，其中非定域性是必不可少的，补充了该领域的最新发展[25，19，11，10，9]。
We consider the classical Yaglom limit theorem for a branching Markov process X = ( X t , t ≥ 0) , with non-local branching mechanism in the setting that the mean semigroup is critical, i.e. its leading eigenvalue is zero. In particular, we show that there exists a constant c ( f ) such that where e c ( f ) is an exponential random variable with rate c ( f ) and the convergence is in distribution. As part of the proof, we also show that the probability of survival decays inversely proportionally to time. Although Yaglom limit theorems have recently been handled in the setting of branching Brownian motion in a bounded domain and superprocesses, [35, 36], these results do not allow for non-local 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 k -th martingale moments of X (rather than the Yaglom limit itself). We then illustrate our results in the setting of neutron transport, for which the non-locality is essential, complementing recent developments in this domain [25, 19, 11, 10, 9].