Mutual service processes in Euclidean spaces: existence and ergodicity

Consider a set of objects, abstracted to points of a spatially stationary point process in $$\mathbb {R}^d$$Rd, 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.

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