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For Cox processes we construct a Markov process with increasing paths tocouple the condensations of the Cox process in a monotone way. A similarprocedure procedure yields an analogue Markov process for the P\'olya sumprocess. Moreover, we identify the exit spaces of these Markov processes andidentify them firstly as mixtures of certain extremal processes, i.e. as aprocess in a random environment, and secondly as Gibbs processes.

Exit spaces for Cox processes and the P\'olya sum process