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Mutual service processes in Euclidean spaces: existence and ergodicity. (English) Zbl 1373.60021

Summary: Consider a set of objects, abstracted to points of a spatially stationary point process in \(\mathbb {R}^d\), that deliver to each other a service at a rate depending on their distance. Assume that the points arrive as a Poisson process and leave when their service requirements have been fulfilled. We show how such a process can be constructed and establish its ergodicity under fairly general conditions. We also establish a hierarchy of integral balance relations between the factorial moment measures and show that the time-stationary process exhibits a repulsivity property.

MSC:

60D05 Geometric probability and stochastic geometry
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G17 Sample path properties
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
05C80 Random graphs (graph-theoretic aspects)
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