分支随机游走的胚单调族的临界参数
Critical parameters of germ-monotone families of branching random walks
AI总结：
提出胚单调分支随机游走（GMBRW）族，定义与子集相关的临界参数，统一并扩展了全局和局部临界参数，并研究繁殖律修改对临界参数的影响。
摘要：
我们在可数集 $X$ 上引入了一类广泛的分支随机游走族，称为胚单调分支随机游走（GMBRWs）。每个族中的过程由正参数 $\lambda>0$ 参数化，该参数控制整体繁殖速度，并且这些过程在胚序（一种扩展经典随机占优的概念）下关于 $\lambda$ 单调递增。该框架涵盖了广泛的模型，包括经典连续时间分支随机游走，以及某些非马尔可夫过程（如老化分支随机游走）的离散时间对应物。我们定义了与每个子集 $A \subseteq X$ 相关的临界参数 $\lambda(A)$ 的一般概念，该参数作为 $A$ 中几乎必然灭绝与 $A$ 中正存活概率之间的阈值。这统一并扩展了经典的全局和局部临界参数 $\lambda_w$ 和 $\lambda_s$，它们可以作为特例恢复。然后，我们研究了繁殖律在有限集或 $X$ 的更一般子集上的修改如何影响这些临界参数。我们的结果扩展了文献中的早期贡献。
英文摘要：
We introduce a broad class of families of branching random walks on a countable set $X$, which we refer to as germ-monotone branching random walks (GMBRWs). The processes in each family are parametrized by a positive parameter $λ>0$, which controls the overall reproductive speed, and they are monotonically increasing in $λ$ with respect to the germ order, a notion that extends classical stochastic domination. This framework encompasses a wide range of models, including classical continuous-time branching random walks, as well as discrete-time counterparts of certain non-Markovian processes such as ageing branching random walks. We define a general notion of critical parameter $λ(A)$ associated with each subset $A \subseteq X$, which serves as a threshold separating almost sure extinction in $A$ from positive probability of survival in $A$. This unifies and extends the classical global and local critical parameters $λ_w$ and $λ_s$, which can be recovered as special cases. We then investigate how modifications of the reproduction laws, either on a finite set or on a more general subset of $X$, affect these critical parameters. Our results extend earlier contributions in the literature.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
