Two representations of a conditioned superprocess
Two representations of a conditioned superprocess
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DOI：
10.1017/s0308210500029619
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发表时间：
1993
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影响因子：
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通讯作者：
S. Evans
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文献类型：
--
作者：
S. Evans
我们考虑一类测度值马尔可夫过程，它是通过在某个底层马尔可夫过程上取一个超过程并使其永远存活而构造的。我们得到这样一个过程的两个表示。第一种表示是一个“不朽的粒子”，它根据潜在的马尔可夫过程运动并抛出质量块，然后以与无条件超过程的质量演化相同的方式继续演化。作为这种表示的结果，我们证明了，如果基础过程的尾σ-域是平凡的，则条件超过程的尾σ-域是平凡的。第二种表示法类似于勒加尔在无条件情况下得到的表示法。它表示条件超过程，条件超过程表示在底层过程的路径空间中取值的某个过程。这种表示对于研究闭支撑过程的“瞬时性”和“常返性”是有用的。
Synopsis We consider a class of measure-valued Markov processes constructed by taking a superprocess over some underlying Markov process and conditioning it to stay alive forever. We obtain two representations of such a process. The first representation is in terms of an “immortal particle” that moves around according to the underlying Markov process and throws off pieces of mass, which then proceed to evolve in the same way that mass evolves for the unconditioned superprocess. As a consequence of this representation, we show that the tail σ-field of the conditioned superprocess is trivial if the tail σ-field of the underlying process is trivial. The second representation is analogous to one obtained by LeGall in the unconditioned case. It represents the conditioned superprocess in terms of a certain process taking values in the path space of the underlying process. This representation is useful for studying the “transience” and “recurrence” properties of the closed support process.