论文标题
通过非标准随机订单进行分支随机步行的瞬变和复发
Transience and recurrence of sets for branching random walk via non-standard stochastic orders
论文作者
论文摘要
我们研究图表上分支随机步行的时空集合的复发和瞬态取决于后代分布。在这里，我们说，如果几乎可以肯定的是，在分支随机步行永远生存的情况下，几乎可以肯定地肯定会肯定地访问时空套装$ a $是经常出现的，并且说如果$ a $几乎肯定会肯定地肯定是有限的。我们证明，如果$ $ $和$ν$是超批评的后代分布，则表示$ \barμ<\barν$，那么相对于后代分布$μ$反复出现的每个时空集，也反复出现，相对于fixpring $ν$，以及与efterient for nimients nsprients相似的$ n $ nspring usprings ncprings n n niments topring n of sspring ncpring n n nive s offing $ n $ $μ$。为了证明这一点，我们引入了有关概率措施的新订单，我们称我们称为细菌订单，并更普遍地证明，只要$μ$在细菌订单中小于$ν$时，相同的结果就会成立。我们的工作灵感来自Johnson and Junge（AIHP 2018）的工作，后者使用相关的随机命令来研究青蛙模型。
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 $μ$ and $ν$ are supercritical offspring distributions with means $\bar μ< \bar ν$ then every space-time set that is recurrent with respect to the offspring distribution $μ$ is also recurrent with respect to the offspring distribution $ν$ and similarly that every space-time set that is transient with respect to the offspring distribution $ν$ is also transient with respect to the offspring distribution $μ$. 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 $μ$ is smaller than $ν$ 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.
