arXiv 1702.035332017-02math.PR

连续状态分支过程的Skelton随机微分方程

Skeletal stochastic differential equations for continuous-state branching process

Dorottya Fekete, Joaquin Fontbona, Andreas E. Kyprianou

备注：: arXiv admin note: text overlap with arXiv:1106.2123 by other authors

摘要：

众所周知，超临界连续态分支过程（CSBP）在规律上等同于离散连续时间Galton Watson过程（多产个体的骨架），其边缘以泊松方式进行修饰，并伴随着移民，从而启动亚临界CSBPs（非多产质量）。在CSBP和超过程的背景下，同样很好理解的是一个以泊松方式打扮的脊柱或不朽粒子的概念，它通过移民启动了原始CSBP的副本，当调节过程以永远生存时，它就会出现。在这篇文章中，我们重新审视这些概念的CSBPs，并把它们放在一个共同的框架使用的语言（耦合）SDEs。通过这种方式，我们能够同时处理所有类型的CSBPs（超临界，临界和亚临界），以及理解骨干表示如何成为弱收敛的意义上，在生存条件下的脊髓分解。我们的主要动机是准备的方式来扩大的空间设置的超过程，最近的结果越来越多地寻求使用骨架分解转移结果从分支粒子设置的措施值过程的设置的方法。

英文摘要：

It is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritial CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and super-processes is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article, we revisit these notions for CSBPs and put them in a common framework using the language of (coupled) SDEs. In this way, we are able to deal simultaneously with all types of CSBPs (supercritical, critical and subcritical) as well as understanding how the backbone representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. Our principal motivation is to prepare the way to expand the SDE approach to the spatial setting of super-processes, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes.

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