Transience and recurrence of sets for branching random walk via non-standard stochastic orders
Transience and recurrence of sets for branching random walk via non-standard stochastic orders
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通过非标准随机顺序进行分支随机游走的集合的瞬态和递归
DOI：
10.1214/21-aihp1186
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Tom Hutchcroft
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文献类型：
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作者：
Tom Hutchcroft
研究了图上分支随机游动时空集的常返性和瞬时性如何依赖于后代分布。在这里，我们说一个时空集合A是递归的，如果它几乎必然地被无限频繁地访问，并且分支随机游走永远存在，并且说A是瞬态的，如果它几乎必然地被最多不频繁地访问。我们证明了如果$\mu$和$\nu$是超临界的后代分布，平均值为$\bar \mu < \bar \nu$，那么每个相对于后代分布$\mu$是常返的时空集也是相对于后代分布$\nu$常返的，同样地，每个相对于后代分布$\nu$是瞬态的时空集也是常返的。对于后代分布$\mu$也是瞬时的。为了证明这一点，我们引入了一个新的顺序的概率措施，我们称之为胚芽的顺序，并证明了更一般的情况下，同样的结果成立时，$\mu$小于$\nu$的胚芽顺序。我们的工作受到了约翰逊和Junge（AIHP 2018）工作的启发，他们使用相关的随机订单来研究青蛙模型。
We study how the recurrence and transience of space-time sets for a branching random walk on a graph depends on the offspring distribution. Here, we say that a space-time set $A$ is recurrent if it is visited infinitely often almost surely on the event that the branching random walk survives forever, and say that $A$ is transient if it is visited at most finitely often almost surely. We prove that if $\mu$ and $\nu$ are supercritical offspring distributions with means $\bar \mu < \bar \nu$ then every space-time set that is recurrent with respect to the offspring distribution $\mu$ is also recurrent with respect to the offspring distribution $\nu$ and similarly that every space-time set that is transient with respect to the offspring distribution $\nu$ is also transient with respect to the offspring distribution $\mu$. To prove this, we introduce a new order on probability measures that we call the germ order and prove more generally that the same result holds whenever $\mu$ is smaller than $\nu$ in the germ order. Our work is inspired by the work of Johnson and Junge (AIHP 2018), who used related stochastic orders to study the frog model.
DOI：
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发表时间：
2014-01
期刊：
arXiv: Probability
影响因子：
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作者：
Elisabetta Candellero;Matthew I. Roberts
通讯作者：
Elisabetta Candellero;Matthew I. Roberts