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